diff --git a/toolchain/check/check.cpp b/toolchain/check/check.cpp
index 786bd8cafdec..5c2e2f2ed253 100644
--- a/toolchain/check/check.cpp
+++ b/toolchain/check/check.cpp
@@ -73,7 +73,8 @@ static auto InitPackageScopeAndImports(Context& context, UnitInfo& unit_info)
     -> void {
   // Define the package scope, with an instruction for `package` expressions to
   // reference.
-  auto package_scope_id = context.name_scopes().Add();
+  auto package_scope_id =
+      context.name_scopes().Add(SemIR::InstId::PackageNamespace);
   CARBON_CHECK(package_scope_id == SemIR::NameScopeId::Package);
 
   auto package_inst = context.AddInst(SemIR::Namespace{
diff --git a/toolchain/check/decl_name_stack.cpp b/toolchain/check/decl_name_stack.cpp
index 669c446217c4..61113601ccca 100644
--- a/toolchain/check/decl_name_stack.cpp
+++ b/toolchain/check/decl_name_stack.cpp
@@ -69,11 +69,22 @@ auto DeclNameStack::LookupOrAddName(NameContext name_context,
         context_->AddNameToLookup(name_context.parse_node,
                                   name_context.unresolved_name_id, target_id);
       } else {
-        // TODO: Reject unless the scope is a namespace scope or the name is
-        // unqualified.
-        bool success = context_->name_scopes().AddEntry(
-            name_context.target_scope_id, name_context.unresolved_name_id,
-            target_id);
+        auto& name_scope =
+            context_->name_scopes().Get(name_context.target_scope_id);
+        if (name_context.has_qualifiers) {
+          auto inst = context_->insts().Get(name_scope.inst_id);
+          if (!inst.Is<SemIR::Namespace>()) {
+            // TODO: Point at the declaration for the scoped entity.
+            CARBON_DIAGNOSTIC(
+                QualifiedDeclOutsideScopeEntity, Error,
+                "Out-of-line declaration requires a declaration in "
+                "scoped entity.");
+            context_->emitter().Emit(name_context.parse_node,
+                                     QualifiedDeclOutsideScopeEntity);
+          }
+        }
+        auto [_, success] = name_scope.names.insert(
+            {name_context.unresolved_name_id, target_id});
         CARBON_CHECK(success)
             << "Duplicate names should have been resolved previously: "
             << name_context.unresolved_name_id << " in "
@@ -102,7 +113,7 @@ auto DeclNameStack::ApplyNameQualifier(Parse::NodeId parse_node,
 auto DeclNameStack::ApplyNameQualifierTo(NameContext& name_context,
                                          Parse::NodeId parse_node,
                                          SemIR::NameId name_id) -> void {
-  if (CanResolveQualifier(name_context, parse_node)) {
+  if (TryResolveQualifier(name_context, parse_node)) {
     // For identifier nodes, we need to perform a lookup on the identifier.
     auto resolved_inst_id = context_->LookupNameInDecl(
         name_context.parse_node, name_id, name_context.target_scope_id);
@@ -152,8 +163,12 @@ auto DeclNameStack::UpdateScopeIfNeeded(NameContext& name_context) -> void {
   }
 }
 
-auto DeclNameStack::CanResolveQualifier(NameContext& name_context,
+auto DeclNameStack::TryResolveQualifier(NameContext& name_context,
                                         Parse::NodeId parse_node) -> bool {
+  // Update has_qualifiers based on the state before any possible changes. If
+  // this is the first qualifier, it may just be the name.
+  name_context.has_qualifiers = name_context.state != NameContext::State::Empty;
+
   switch (name_context.state) {
     case NameContext::State::Error:
       // Already in an error state, so return without examining.
diff --git a/toolchain/check/decl_name_stack.h b/toolchain/check/decl_name_stack.h
index d98b200e3776..a56a4f039f58 100644
--- a/toolchain/check/decl_name_stack.h
+++ b/toolchain/check/decl_name_stack.h
@@ -110,6 +110,9 @@ class DeclNameStack {
 
     State state = State::Empty;
 
+    // Whether there have been qualifiers in the name.
+    bool has_qualifiers = false;
+
     // The scope which qualified names are added to. For unqualified names in
     // an unnamed scope, this will be Invalid to indicate the current scope
     // should be used.
@@ -182,7 +185,7 @@ class DeclNameStack {
 
   // Returns true if the context is in a state where it can resolve qualifiers.
   // Updates name_context as needed.
-  auto CanResolveQualifier(NameContext& name_context, Parse::NodeId parse_node)
+  auto TryResolveQualifier(NameContext& name_context, Parse::NodeId parse_node)
       -> bool;
 
   // Updates the scope on name_context as needed. This is called after
diff --git a/toolchain/check/handle_class.cpp b/toolchain/check/handle_class.cpp
index eedec4710560..948ea3190827 100644
--- a/toolchain/check/handle_class.cpp
+++ b/toolchain/check/handle_class.cpp
@@ -136,7 +136,7 @@ auto HandleClassDefinitionStart(Context& context, Parse::NodeId parse_node)
         .Emit();
   } else {
     class_info.definition_id = class_decl_id;
-    class_info.scope_id = context.name_scopes().Add();
+    class_info.scope_id = context.name_scopes().Add(class_decl_id);
   }
 
   // Enter the class scope.
diff --git a/toolchain/check/handle_interface.cpp b/toolchain/check/handle_interface.cpp
index 9dd1b7256847..e0feeb794045 100644
--- a/toolchain/check/handle_interface.cpp
+++ b/toolchain/check/handle_interface.cpp
@@ -109,7 +109,7 @@ auto HandleInterfaceDefinitionStart(Context& context, Parse::NodeId parse_node)
         .Emit();
   } else {
     interface_info.definition_id = interface_decl_id;
-    interface_info.scope_id = context.name_scopes().Add();
+    interface_info.scope_id = context.name_scopes().Add(interface_decl_id);
   }
 
   // Enter the interface scope.
diff --git a/toolchain/check/handle_namespace.cpp b/toolchain/check/handle_namespace.cpp
index 4445dd27e41e..ebeced48b506 100644
--- a/toolchain/check/handle_namespace.cpp
+++ b/toolchain/check/handle_namespace.cpp
@@ -5,6 +5,7 @@
 #include "toolchain/check/context.h"
 #include "toolchain/check/decl_state.h"
 #include "toolchain/check/modifiers.h"
+#include "toolchain/sem_ir/ids.h"
 #include "toolchain/sem_ir/inst.h"
 
 namespace Carbon::Check {
@@ -21,9 +22,12 @@ auto HandleNamespace(Context& context, Parse::NodeId parse_node) -> bool {
   auto name_context = context.decl_name_stack().FinishName();
   LimitModifiersOnDecl(context, KeywordModifierSet::None,
                        Lex::TokenKind::Namespace);
-  auto namespace_id = context.AddInst(SemIR::Namespace{
+  auto namespace_inst = SemIR::Namespace{
       parse_node, context.GetBuiltinType(SemIR::BuiltinKind::NamespaceType),
-      context.name_scopes().Add()});
+      SemIR::NameScopeId::Invalid};
+  auto namespace_id = context.AddInst(namespace_inst);
+  namespace_inst.name_scope_id = context.name_scopes().Add(namespace_id);
+  context.insts().Set(namespace_id, namespace_inst);
   context.decl_name_stack().AddNameToLookup(name_context, namespace_id);
 
   context.decl_name_stack().PopScope();
diff --git a/toolchain/check/testdata/class/fail_base_method_define.carbon b/toolchain/check/testdata/class/fail_base_method_define.carbon
index db6fb70a196d..d186fe7aa990 100644
--- a/toolchain/check/testdata/class/fail_base_method_define.carbon
+++ b/toolchain/check/testdata/class/fail_base_method_define.carbon
@@ -16,7 +16,9 @@ class D {
   extend base: B;
 }
 
-// TODO: This should be rejected.
+// CHECK:STDERR: fail_base_method_define.carbon:[[@LINE+3]]:6: ERROR: Out-of-line declaration requires a declaration in scoped entity.
+// CHECK:STDERR: fn D.F() {}
+// CHECK:STDERR:      ^
 fn D.F() {}
 
 // CHECK:STDERR: fail_base_method_define.carbon:[[@LINE+3]]:6: ERROR: Name `C` not found.
@@ -40,7 +42,7 @@ fn D.C.F() {}
 // CHECK:STDOUT:   %D.decl = class_decl @D, ()
 // CHECK:STDOUT:   %D: type = class_type @D
 // CHECK:STDOUT:   %F: <function> = fn_decl @F.3
-// CHECK:STDOUT:   %.loc25: <function> = fn_decl @.1
+// CHECK:STDOUT:   %.loc27: <function> = fn_decl @.1
 // CHECK:STDOUT: }
 // CHECK:STDOUT:
 // CHECK:STDOUT: class @B {
diff --git a/toolchain/check/testdata/class/fail_out_of_line_decl.carbon b/toolchain/check/testdata/class/fail_out_of_line_decl.carbon
new file mode 100644
index 000000000000..627409e0eca1
--- /dev/null
+++ b/toolchain/check/testdata/class/fail_out_of_line_decl.carbon
@@ -0,0 +1,37 @@
+// Part of the Carbon Language project, under the Apache License v2.0 with LLVM
+// Exceptions. See /LICENSE for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+// AUTOUPDATE
+
+class C {}
+
+// CHECK:STDERR: fail_out_of_line_decl.carbon:[[@LINE+3]]:6: ERROR: Out-of-line declaration requires a declaration in scoped entity.
+// CHECK:STDERR: fn C.F() {}
+// CHECK:STDERR:      ^
+fn C.F() {}
+
+// CHECK:STDOUT: --- fail_out_of_line_decl.carbon
+// CHECK:STDOUT:
+// CHECK:STDOUT: constants {
+// CHECK:STDOUT:   %.loc7: type = struct_type {}
+// CHECK:STDOUT: }
+// CHECK:STDOUT:
+// CHECK:STDOUT: file {
+// CHECK:STDOUT:   package: <namespace> = namespace {.C = %C.decl}
+// CHECK:STDOUT:   %C.decl = class_decl @C, ()
+// CHECK:STDOUT:   %C: type = class_type @C
+// CHECK:STDOUT:   %F: <function> = fn_decl @F
+// CHECK:STDOUT: }
+// CHECK:STDOUT:
+// CHECK:STDOUT: class @C {
+// CHECK:STDOUT:
+// CHECK:STDOUT: !members:
+// CHECK:STDOUT:   .F = file.%F
+// CHECK:STDOUT: }
+// CHECK:STDOUT:
+// CHECK:STDOUT: fn @F() {
+// CHECK:STDOUT: !entry:
+// CHECK:STDOUT:   return
+// CHECK:STDOUT: }
+// CHECK:STDOUT:
diff --git a/toolchain/diagnostics/diagnostic_kind.def b/toolchain/diagnostics/diagnostic_kind.def
index 52cce33e09f8..0828219bf201 100644
--- a/toolchain/diagnostics/diagnostic_kind.def
+++ b/toolchain/diagnostics/diagnostic_kind.def
@@ -214,6 +214,7 @@ CARBON_DIAGNOSTIC_KIND(ReturnStatementMissingExpr)
 CARBON_DIAGNOSTIC_KIND(ImplicitAsConversionFailure)
 CARBON_DIAGNOSTIC_KIND(ExplicitAsConversionFailure)
 CARBON_DIAGNOSTIC_KIND(TypeExprEvaluationFailure)
+CARBON_DIAGNOSTIC_KIND(QualifiedDeclOutsideScopeEntity)
 CARBON_DIAGNOSTIC_KIND(QualifiedDeclInIncompleteClassScope)
 CARBON_DIAGNOSTIC_KIND(QualifiedDeclInNonScope)
 CARBON_DIAGNOSTIC_KIND(QualifiedDeclNonScopeEntity)
diff --git a/toolchain/sem_ir/value_stores.h b/toolchain/sem_ir/value_stores.h
index aaef8b951e4b..d1d353305ff5 100644
--- a/toolchain/sem_ir/value_stores.h
+++ b/toolchain/sem_ir/value_stores.h
@@ -173,7 +173,7 @@ class NameStoreWrapper {
 
 struct NameScope {
   // Names in the scope.
-  llvm::DenseMap<NameId, InstId> names;
+  llvm::DenseMap<NameId, InstId> names = llvm::DenseMap<NameId, InstId>();
 
   // Scopes extended by this scope.
   //
@@ -193,6 +193,9 @@ struct NameScope {
   // TODO: Consider using something like `TinyPtrVector` for this.
   llvm::SmallVector<NameScopeId, 1> extended_scopes;
 
+  // The instructioning which owns the scope.
+  InstId inst_id;
+
   // Whether we have diagnosed an error in a construct that would have added
   // names to this scope. For example, this can happen if an `import` failed or
   // an `extend` declaration was ill-formed. If true, the `names` map is
@@ -205,7 +208,9 @@ struct NameScope {
 class NameScopeStore {
  public:
   // Adds a name scope, returning an ID to reference it.
-  auto Add() -> NameScopeId { return values_.AddDefaultValue(); }
+  auto Add(InstId inst_id) -> NameScopeId {
+    return values_.Add({.inst_id = inst_id});
+  }
 
   // Adds an entry to a name scope. Returns true on success, false on
   // duplicates.