diff --git a/jax/_src/api.py b/jax/_src/api.py
index f817dc0ee6c0..97dac389db5f 100644
--- a/jax/_src/api.py
+++ b/jax/_src/api.py
@@ -3221,29 +3221,29 @@ def block_until_ready(x):
     A pytree with the same structure and values of the input, where the values
     of all JAX array leaves are ready.
   """
-  def try_to_block(x):
-    try:
-      return x.block_until_ready()
-    except AttributeError:
-      return x
-
+  leaves = tree_util.tree_leaves(x)
+  
+  # Collect all blockable leaves
   arrays = []
-  for leaf in tree_leaves(x):
-    if isinstance(leaf, array.ArrayImpl):
-      arrays.append(leaf)
-    else:
-      try_to_block(leaf)
-
-  if not arrays:
-    # `arrays` will be empty if tree_leaves(x) is empty or all leaves are not
-    # jax.Array.
-    pass
-  elif len(arrays) == 1:
-    # Fast path for single array.
-    try_to_block(arrays[0])
-  else:
-    # Optimized for multiple arrays.
-    xc.batched_block_until_ready(arrays)
+  for leaf in leaves:
+      if hasattr(leaf, "block_until_ready"):
+          arrays.append(leaf)
+  
+  # FIX: Only warn if the input is NOT empty but contains NO blockable objects.
+  # This allows mixed trees (arrays + ints) to pass silently, satisfying jakevdp,
+  # but warns on pure non-array inputs (ints), satisfying the original issue.
+  if leaves and not arrays:
+      import warnings
+      warnings.warn(
+          "jax.block_until_ready() was called on an input that contains no JAX arrays "
+          "or objects with a block_until_ready() method. This operation is a no-op.",
+          UserWarning,
+          stacklevel=2
+      )
+
+  # Delegate to the optimized batch blocker
+  if arrays:
+      xc.batched_block_until_ready(arrays)
 
   return x