diff --git a/src/runtime/time.go b/src/runtime/time.go
index 83d93c5686..d338705b7c 100644
--- a/src/runtime/time.go
+++ b/src/runtime/time.go
@@ -187,6 +187,9 @@ func timeSleep(ns int64) {
 	t.f = goroutineReady
 	t.arg = gp
 	t.nextwhen = nanotime() + ns
+	if t.nextwhen < 0 { // check for overflow.
+		t.nextwhen = maxWhen
+	}
 	gopark(resetForSleep, unsafe.Pointer(t), waitReasonSleep, traceEvGoSleep, 1)
 }
 
@@ -244,10 +247,14 @@ func goroutineReady(arg interface{}, seq uintptr) {
 // That avoids the risk of changing the when field of a timer in some P's heap,
 // which could cause the heap to become unsorted.
 func addtimer(t *timer) {
-	// when must never be negative; otherwise runtimer will overflow
-	// during its delta calculation and never expire other runtime timers.
-	if t.when < 0 {
-		t.when = maxWhen
+	// when must be positive. A negative value will cause runtimer to
+	// overflow during its delta calculation and never expire other runtime
+	// timers. Zero will cause checkTimers to fail to notice the timer.
+	if t.when <= 0 {
+		throw("timer when must be positive")
+	}
+	if t.period < 0 {
+		throw("timer period must be non-negative")
 	}
 	if t.status != timerNoStatus {
 		throw("addtimer called with initialized timer")
@@ -408,8 +415,11 @@ func dodeltimer0(pp *p) {
 // This is called by the netpoll code or time.Ticker.Reset or time.Timer.Reset.
 // Reports whether the timer was modified before it was run.
 func modtimer(t *timer, when, period int64, f func(interface{}, uintptr), arg interface{}, seq uintptr) bool {
-	if when < 0 {
-		when = maxWhen
+	if when <= 0 {
+		throw("timer when must be positive")
+	}
+	if period < 0 {
+		throw("timer period must be non-negative")
 	}
 
 	status := uint32(timerNoStatus)
@@ -848,6 +858,9 @@ func runOneTimer(pp *p, t *timer, now int64) {
 		// Leave in heap but adjust next time to fire.
 		delta := t.when - now
 		t.when += t.period * (1 + -delta/t.period)
+		if t.when < 0 { // check for overflow.
+			t.when = maxWhen
+		}
 		siftdownTimer(pp.timers, 0)
 		if !atomic.Cas(&t.status, timerRunning, timerWaiting) {
 			badTimer()
@@ -1066,6 +1079,9 @@ func siftupTimer(t []*timer, i int) {
 		badTimer()
 	}
 	when := t[i].when
+	if when <= 0 {
+		badTimer()
+	}
 	tmp := t[i]
 	for i > 0 {
 		p := (i - 1) / 4 // parent
@@ -1086,6 +1102,9 @@ func siftdownTimer(t []*timer, i int) {
 		badTimer()
 	}
 	when := t[i].when
+	if when <= 0 {
+		badTimer()
+	}
 	tmp := t[i]
 	for {
 		c := i*4 + 1 // left child
diff --git a/src/time/internal_test.go b/src/time/internal_test.go
index e70b6f34de..ffe54e47c2 100644
--- a/src/time/internal_test.go
+++ b/src/time/internal_test.go
@@ -33,38 +33,30 @@ var DaysIn = daysIn
 
 func empty(arg interface{}, seq uintptr) {}
 
-// Test that a runtimeTimer with a duration so large it overflows
-// does not cause other timers to hang.
+// Test that a runtimeTimer with a period that would overflow when on
+// expiration does not throw or cause other timers to hang.
 //
 // This test has to be in internal_test.go since it fiddles with
 // unexported data structures.
-func CheckRuntimeTimerOverflow() {
-	// We manually create a runtimeTimer to bypass the overflow
-	// detection logic in NewTimer: we're testing the underlying
-	// runtime.addtimer function.
+func CheckRuntimeTimerPeriodOverflow() {
+	// We manually create a runtimeTimer with huge period, but that expires
+	// immediately. The public Timer interface would require waiting for
+	// the entire period before the first update.
 	r := &runtimeTimer{
-		when: runtimeNano() + (1<<63 - 1),
-		f:    empty,
-		arg:  nil,
+		when:   runtimeNano(),
+		period: 1<<63 - 1,
+		f:      empty,
+		arg:    nil,
 	}
 	startTimer(r)
+	defer stopTimer(r)
 
-	// Start a goroutine that should send on t.C right away.
-	t := NewTimer(1)
-
-	defer func() {
-		stopTimer(r)
-		t.Stop()
-	}()
-
-	// If the test fails, we will hang here until the timeout in the
-	// testing package fires, which is 10 minutes. It would be nice to
-	// catch the problem sooner, but there is no reliable way to guarantee
-	// that timers are run without doing something involving the scheduler.
-	// Previous failed attempts have tried calling runtime.Gosched and
-	// runtime.GC, but neither is reliable. So we fall back to hope:
-	// We hope we don't hang here.
-	<-t.C
+	// If this test fails, we will either throw (when siftdownTimer detects
+	// bad when on update), or other timers will hang (if the timer in a
+	// heap is in a bad state). There is no reliable way to test this, but
+	// we wait on a short timer here as a smoke test (alternatively, timers
+	// in later tests may hang).
+	<-After(25 * Millisecond)
 }
 
 var (
diff --git a/src/time/sleep.go b/src/time/sleep.go
index 22ffd68282..90d8a18a68 100644
--- a/src/time/sleep.go
+++ b/src/time/sleep.go
@@ -31,6 +31,8 @@ func when(d Duration) int64 {
 	}
 	t := runtimeNano() + int64(d)
 	if t < 0 {
+		// N.B. runtimeNano() and d are always positive, so addition
+		// (including overflow) will never result in t == 0.
 		t = 1<<63 - 1 // math.MaxInt64
 	}
 	return t
diff --git a/src/time/sleep_test.go b/src/time/sleep_test.go
index ba0016bf49..084ac33f51 100644
--- a/src/time/sleep_test.go
+++ b/src/time/sleep_test.go
@@ -434,17 +434,29 @@ func TestReset(t *testing.T) {
 	t.Error(err)
 }
 
-// Test that sleeping for an interval so large it overflows does not
-// result in a short sleep duration.
+// Test that sleeping (via Sleep or Timer) for an interval so large it
+// overflows does not result in a short sleep duration. Nor does it interfere
+// with execution of other timers. If it does, timers in this or subsequent
+// tests may not fire.
 func TestOverflowSleep(t *testing.T) {
 	const big = Duration(int64(1<<63 - 1))
+
+	go func() {
+		Sleep(big)
+		// On failure, this may return after the test has completed, so
+		// we need to panic instead.
+		panic("big sleep returned")
+	}()
+
 	select {
 	case <-After(big):
 		t.Fatalf("big timeout fired")
 	case <-After(25 * Millisecond):
 		// OK
 	}
+
 	const neg = Duration(-1 << 63)
+	Sleep(neg) // Returns immediately.
 	select {
 	case <-After(neg):
 		// OK
@@ -473,13 +485,10 @@ func TestIssue5745(t *testing.T) {
 	t.Error("Should be unreachable.")
 }
 
-func TestOverflowRuntimeTimer(t *testing.T) {
-	if testing.Short() {
-		t.Skip("skipping in short mode, see issue 6874")
-	}
+func TestOverflowPeriodRuntimeTimer(t *testing.T) {
 	// This may hang forever if timers are broken. See comment near
 	// the end of CheckRuntimeTimerOverflow in internal_test.go.
-	CheckRuntimeTimerOverflow()
+	CheckRuntimeTimerPeriodOverflow()
 }
 
 func checkZeroPanicString(t *testing.T) {