diff --git a/pkg/ccl/logictestccl/tests/3node-tenant/generated_test.go b/pkg/ccl/logictestccl/tests/3node-tenant/generated_test.go
index ae76ecf9c9d1..ffa117e87bd8 100644
--- a/pkg/ccl/logictestccl/tests/3node-tenant/generated_test.go
+++ b/pkg/ccl/logictestccl/tests/3node-tenant/generated_test.go
@@ -81,6 +81,9 @@ func TestMain(m *testing.M) {
 }
 
 func runLogicTest(t *testing.T, file string) {
+	if file == "fk" || file == "alter_primary_key" {
+		skip.UnderRace(t, "times out and/or OOM's")
+	}
 	skip.UnderDeadlock(t, "times out and/or hangs")
 	logictest.RunLogicTest(t, logictest.TestServerArgs{}, configIdx, filepath.Join(logicTestDir, file))
 }
diff --git a/pkg/cmd/generate-logictest/main.go b/pkg/cmd/generate-logictest/main.go
index 8b64ebf880ed..cad366cc2d34 100644
--- a/pkg/cmd/generate-logictest/main.go
+++ b/pkg/cmd/generate-logictest/main.go
@@ -33,6 +33,7 @@ type testFileTemplateConfig struct {
 	Ccl                           bool
 	ForceProductionValues         bool
 	IsMultiRegion                 bool
+	Is3NodeTenant                 bool
 	Package, TestRuleName, RelDir string
 	ConfigIdx                     int
 	TestCount                     int
@@ -176,6 +177,7 @@ func (t *testdir) dump() error {
 		tplCfg.TestCount = testCount
 		tplCfg.CockroachGoTestserverTest = cfg.UseCockroachGoTestserver
 		tplCfg.IsMultiRegion = strings.Contains(cfg.Name, "multiregion")
+		tplCfg.Is3NodeTenant = strings.HasSuffix(cfg.Name, "3node-tenant")
 		// The NumCPU calculation is a guess pulled out of thin air to
 		// allocate the tests which use 3-node clusters 2 vCPUs, and
 		// the ones which use more a bit more.
diff --git a/pkg/cmd/generate-logictest/templates.go b/pkg/cmd/generate-logictest/templates.go
index 26208744f389..220737926254 100644
--- a/pkg/cmd/generate-logictest/templates.go
+++ b/pkg/cmd/generate-logictest/templates.go
@@ -51,7 +51,10 @@ const templateText = `
 {{- define "runLogicTest" }}
 {{- if .LogicTest -}}
 func runLogicTest(t *testing.T, file string) {
-	skip.UnderDeadlock(t, "times out and/or hangs")
+	{{ if .Is3NodeTenant }}if file == "fk" || file == "alter_primary_key" {
+		skip.UnderRace(t, "times out and/or OOM's")
+	}
+	{{ end }}skip.UnderDeadlock(t, "times out and/or hangs")
 	logictest.RunLogicTest(t, logictest.TestServerArgs{}, configIdx, filepath.Join(logicTestDir, file))
 }
 {{ end }}
diff --git a/pkg/server/application_api/sql_stats_test.go b/pkg/server/application_api/sql_stats_test.go
index 738f05310c49..2bdf538a6855 100644
--- a/pkg/server/application_api/sql_stats_test.go
+++ b/pkg/server/application_api/sql_stats_test.go
@@ -55,6 +55,8 @@ func TestStatusAPICombinedTransactions(t *testing.T) {
 	defer leaktest.AfterTest(t)()
 	defer log.Scope(t).Close(t)
 
+	skip.UnderRace(t, "probable OOM")
+
 	// Increase the timeout for the http client if under stress.
 	additionalTimeout := 0 * time.Second
 	if skip.Stress() {