diff --git a/promql/engine.go b/promql/engine.go
index 4a894fe72..161aa85ac 100644
--- a/promql/engine.go
+++ b/promql/engine.go
@@ -60,6 +60,11 @@ const (
 	maxInt64 = 9223372036854774784
 	// The smallest SampleValue that can be converted to an int64 without underflow.
 	minInt64 = -9223372036854775808
+
+	// Max initial size for the pooled points slices.
+	// The getHPointSlice and getFPointSlice functions are called with an estimated size which often can be
+	// over-estimated.
+	maxPointsSliceSize = 5000
 )
 
 type engineMetrics struct {
@@ -1910,19 +1915,33 @@ func getFPointSlice(sz int) []FPoint {
 	if p := fPointPool.Get(); p != nil {
 		return p
 	}
+
+	if sz > maxPointsSliceSize {
+		sz = maxPointsSliceSize
+	}
+
 	return make([]FPoint, 0, sz)
 }
 
+// putFPointSlice will return a FPoint slice of size max(maxPointsSliceSize, sz).
+// This function is called with an estimated size which often can be over-estimated.
 func putFPointSlice(p []FPoint) {
 	if p != nil {
 		fPointPool.Put(p[:0])
 	}
 }
 
+// getHPointSlice will return a HPoint slice of size max(maxPointsSliceSize, sz).
+// This function is called with an estimated size which often can be over-estimated.
 func getHPointSlice(sz int) []HPoint {
 	if p := hPointPool.Get(); p != nil {
 		return p
 	}
+
+	if sz > maxPointsSliceSize {
+		sz = maxPointsSliceSize
+	}
+
 	return make([]HPoint, 0, sz)
 }