diff --git a/lib/rspec/matchers/built_in/start_or_end_with.rb b/lib/rspec/matchers/built_in/start_or_end_with.rb
index 81f06c288..53a22ebc1 100644
--- a/lib/rspec/matchers/built_in/start_or_end_with.rb
+++ b/lib/rspec/matchers/built_in/start_or_end_with.rb
@@ -13,11 +13,12 @@ def initialize(*expected)
         # @api private
         # @return [String]
         def failure_message
+          response_msg = ", but it does not respond to #{method} and cannot be indexed using #[]"
           super.tap do |msg|
             if @actual_does_not_have_ordered_elements
               msg << ", but it does not have ordered elements"
             elsif !actual.respond_to?(:[])
-              msg << ", but it cannot be indexed using #[]"
+              msg << response_msg
             end
           end
         end
@@ -33,7 +34,9 @@ def description
 
       private
 
-        def match(_expected, actual)
+        def match(expected, actual)
+          # use an object's start_with? or end_with? as appropriate
+          return actual.send(method, expected) if actual.respond_to?(method)
           return false unless actual.respond_to?(:[])
 
           begin
@@ -73,6 +76,10 @@ def subset_matches?
         def element_matches?
           values_match?(expected, actual[0])
         end
+
+        def method
+          :start_with?
+        end
       end
 
       # @api private
@@ -88,6 +95,10 @@ def subset_matches?
         def element_matches?
           values_match?(expected, actual[-1])
         end
+
+        def method
+          :end_with?
+        end
       end
     end
   end
diff --git a/spec/rspec/matchers/built_in/start_and_end_with_spec.rb b/spec/rspec/matchers/built_in/start_and_end_with_spec.rb
index b063f199d..02bed68c9 100644
--- a/spec/rspec/matchers/built_in/start_and_end_with_spec.rb
+++ b/spec/rspec/matchers/built_in/start_and_end_with_spec.rb
@@ -102,11 +102,25 @@ def ==(other)
   end
 
   context "with an object that does not respond to :[]" do
-    it "fails with a useful message" do
-      actual = Object.new
-      expect {
-        expect(actual).to start_with 0
-      }.to fail_with("expected #{actual.inspect} to start with 0, but it cannot be indexed using #[]")
+    context "with an object that responds to start_with?" do
+      it "relies on start_with?" do
+        my_struct = Struct.new(:foo) do
+          def start_with?(elem)
+            true
+          end
+        end
+
+        expect(my_struct.new("foo")).to start_with(0)
+      end
+    end
+
+    context "with an object that does not respond to start_with?" do
+      it "fails with a useful message" do
+        actual = Object.new
+        expect {
+          expect(actual).to start_with 0
+        }.to fail_with("expected #{actual.inspect} to start with 0, but it does not respond to start_with? and cannot be indexed using #[]")
+      end
     end
   end
 
@@ -310,11 +324,24 @@ def ==(other)
   end
 
   context "with an object that does not respond to :[]" do
-    it "fails with a useful message" do
-      actual = Object.new
-      expect {
-        expect(actual).to end_with 0
-      }.to fail_with("expected #{actual.inspect} to end with 0, but it cannot be indexed using #[]")
+    context "with an object that responds to end_with?" do
+      it "relies on end_with?" do
+        my_struct = Struct.new(:foo) do
+          def end_with?(elem)
+            true
+          end
+        end
+        expect(my_struct.new("foo")).to end_with(0)
+      end
+    end
+
+    context "with an object that does not respond to end_with?" do
+      it "fails with a useful message" do
+        actual = Object.new
+        expect {
+          expect(actual).to end_with 0
+        }.to fail_with("expected #{actual.inspect} to end with 0, but it does not respond to end_with? and cannot be indexed using #[]")
+      end
     end
   end