diff --git a/doc/src/sgml/monitoring.sgml b/doc/src/sgml/monitoring.sgml
index dcf6e6a2f48..2aeb38a6e5f 100644
--- a/doc/src/sgml/monitoring.sgml
+++ b/doc/src/sgml/monitoring.sgml
@@ -6265,10 +6265,13 @@ FROM pg_stat_get_backend_idset() AS backendid;
        <structfield>tuples_skipped</structfield> <type>bigint</type>
       </para>
       <para>
-       Number of tuples skipped because they contain malformed data.
-       This counter only advances when
-       <literal>ignore</literal> is specified to the <literal>ON_ERROR</literal>
-       option.
+       Number of tuples that contained malformed data. When
+       <literal>ON_ERROR</literal> is set to <literal>ignore</literal>,
+       this counts rows that were skipped. When set to
+       <literal>set_null</literal>, this counts rows where at least one
+       column was set to null due to a conversion error.
+       This counter only advances when <literal>ON_ERROR</literal>
+       is set to <literal>ignore</literal> or <literal>set_null</literal>.
       </para></entry>
      </row>
     </tbody>
diff --git a/src/backend/commands/copyfromparse.c b/src/backend/commands/copyfromparse.c
index 96ba23e961c..0a75800b8ab 100644
--- a/src/backend/commands/copyfromparse.c
+++ b/src/backend/commands/copyfromparse.c
@@ -1052,6 +1052,11 @@ CopyFromTextLikeOneRow(CopyFromState cstate, ExprContext *econtext,
 				cstate->num_errors++;
 			else if (cstate->opts.on_error == COPY_ON_ERROR_SET_NULL)
 			{
+				/*
+				 * Reset error state so the subsequent InputFunctionCallSafe
+				 * call (for domain constraint check) can properly report
+				 * whether it succeeded or failed.
+				 */
 				cstate->escontext->error_occurred = false;
 
 				Assert(cstate->domain_with_constraint != NULL);
@@ -1075,13 +1080,17 @@ CopyFromTextLikeOneRow(CopyFromState cstate, ExprContext *econtext,
 				else if (string == NULL)
 					ereport(ERROR,
 							errcode(ERRCODE_NOT_NULL_VIOLATION),
-							errmsg("domain %s does not allow null values", format_type_be(typioparams[m])),
+							errmsg("null value in column \"%s\" violates not-null constraint of domain %s",
+								   cstate->cur_attname, format_type_be(typioparams[m])),
 							errdatatype(typioparams[m]));
 				else
 					ereport(ERROR,
-							errcode(ERRCODE_INVALID_TEXT_REPRESENTATION),
-							errmsg("invalid input value for domain %s: \"%s\"",
-								   format_type_be(typioparams[m]), string));
+							errcode(ERRCODE_NOT_NULL_VIOLATION),
+							errmsg("cannot set null value for column \"%s\" with domain %s",
+								   cstate->cur_attname, format_type_be(typioparams[m])),
+							errdetail("Column \"%s\" does not accept null values, so ON_ERROR SET_NULL cannot be applied.",
+									  cstate->cur_attname),
+							errdatatype(typioparams[m]));
 
 				/*
 				 * We count only the number of rows (not fields) where
diff --git a/src/test/regress/expected/copy2.out b/src/test/regress/expected/copy2.out
index 72034796aca..d9cc7bf5f48 100644
--- a/src/test/regress/expected/copy2.out
+++ b/src/test/regress/expected/copy2.out
@@ -799,13 +799,15 @@ CREATE DOMAIN d_int_positive_maybe_null AS integer CHECK(value > 0);
 CREATE TABLE t_on_error_null (a d_int_not_null, b d_int_positive_maybe_null, c integer);
 \pset null NULL
 COPY t_on_error_null FROM STDIN WITH (on_error set_null); --fail
-ERROR:  domain d_int_not_null does not allow null values
+ERROR:  null value in column "a" violates not-null constraint of domain d_int_not_null
 CONTEXT:  COPY t_on_error_null, line 1, column a: null input
 COPY t_on_error_null FROM STDIN WITH (on_error set_null); --fail
-ERROR:  invalid input value for domain d_int_not_null: "ss"
+ERROR:  cannot set null value for column "a" with domain d_int_not_null
+DETAIL:  Column "a" does not accept null values, so ON_ERROR SET_NULL cannot be applied.
 CONTEXT:  COPY t_on_error_null, line 1, column a: "ss"
 COPY t_on_error_null FROM STDIN WITH (on_error set_null); --fail
-ERROR:  invalid input value for domain d_int_not_null: "-1"
+ERROR:  cannot set null value for column "a" with domain d_int_not_null
+DETAIL:  Column "a" does not accept null values, so ON_ERROR SET_NULL cannot be applied.
 CONTEXT:  COPY t_on_error_null, line 1, column a: "-1"
 --fail, less data.
 COPY t_on_error_null FROM STDIN WITH (delimiter ',', on_error set_null);