diff --git a/lucene/CHANGES.txt b/lucene/CHANGES.txt
index 467f697f051..0359a8e8543 100644
--- a/lucene/CHANGES.txt
+++ b/lucene/CHANGES.txt
@@ -139,6 +139,9 @@ Optimizations
   computes scores lazily in order to be faster when used in conjunctions.
   (Adrien Grand)
 
+* LUCENE-7667: BKDReader now calls `IntersectVisitor.grow()` on larger
+  increments. (Adrien Grand)
+
 Build
 
 * LUCENE-7651: Fix Javadocs build for Java 8u121 by injecting "Google Code
diff --git a/lucene/core/src/java/org/apache/lucene/index/PointValues.java b/lucene/core/src/java/org/apache/lucene/index/PointValues.java
index 01f77e46509..dab91403a18 100644
--- a/lucene/core/src/java/org/apache/lucene/index/PointValues.java
+++ b/lucene/core/src/java/org/apache/lucene/index/PointValues.java
@@ -211,8 +211,7 @@ public abstract class PointValues {
      *  determine how to further recurse down the tree. */
     Relation compare(byte[] minPackedValue, byte[] maxPackedValue);
 
-    /** Notifies the caller that this many documents (from one block) are about
-     *  to be visited */
+    /** Notifies the caller that this many documents are about to be visited */
     default void grow(int count) {};
   }
 
diff --git a/lucene/core/src/java/org/apache/lucene/util/bkd/BKDReader.java b/lucene/core/src/java/org/apache/lucene/util/bkd/BKDReader.java
index e1204358f82..ad659907a8e 100644
--- a/lucene/core/src/java/org/apache/lucene/util/bkd/BKDReader.java
+++ b/lucene/core/src/java/org/apache/lucene/util/bkd/BKDReader.java
@@ -528,10 +528,19 @@ public final class BKDReader extends PointValues implements Accountable {
   }
 
   /** Fast path: this is called when the query box fully encompasses all cells under this node. */
-  private void addAll(IntersectState state) throws IOException {
+  private void addAll(IntersectState state, boolean grown) throws IOException {
     //System.out.println("R: addAll nodeID=" + nodeID);
 
+    if (grown == false) {
+      final long maxPointCount = (long) maxPointsInLeafNode * state.index.getNumLeaves();
+      if (maxPointCount <= Integer.MAX_VALUE) { // could be >MAX_VALUE if there are more than 2B points in total
+        state.visitor.grow((int) maxPointCount);
+        grown = true;
+      }
+    }
+
     if (state.index.isLeafNode()) {
+      assert grown;
       //System.out.println("ADDALL");
       if (state.index.nodeExists()) {
         visitDocIDs(state.in, state.index.getLeafBlockFP(), state.visitor);
@@ -539,11 +548,11 @@ public final class BKDReader extends PointValues implements Accountable {
       // TODO: we can assert that the first value here in fact matches what the index claimed?
     } else {
       state.index.pushLeft();
-      addAll(state);
+      addAll(state, grown);
       state.index.pop();
 
       state.index.pushRight();
-      addAll(state);
+      addAll(state, grown);
       state.index.pop();
     }
   }
@@ -579,7 +588,7 @@ public final class BKDReader extends PointValues implements Accountable {
 
     // How many points are stored in this leaf cell:
     int count = in.readVInt();
-    visitor.grow(count);
+    // No need to call grow(), it has been called up-front
 
     if (version < BKDWriter.VERSION_COMPRESSED_DOC_IDS) {
       DocIdsWriter.readInts32(in, count, visitor);
@@ -687,7 +696,7 @@ public final class BKDReader extends PointValues implements Accountable {
       // This cell is fully outside of the query shape: stop recursing
     } else if (r == Relation.CELL_INSIDE_QUERY) {
       // This cell is fully inside of the query shape: recursively add all points in this cell without filtering
-      addAll(state);
+      addAll(state, false);
       // The cell crosses the shape boundary, or the cell fully contains the query, so we fall through and do full filtering:
     } else if (state.index.isLeafNode()) {