This might be a little tricky to reproduce, but this seems to work rather reliably:

• Put down an anvil and get a stack of some type of item.
• Insert a single item into the anvil and rename it, for ex. "A".
• Pickup the item and then drag it (right-mouse button) across slots inside your inventory.
• Insert another item from the stack into the anvil and rename it to "A" as well.
• Now carefully pickup that item from the anvil (with a single left click), and place it on top of the previously renamed item: Make sure to not cause 'inventory dragging' when doing this.
• The items will seem to stack.
• Close the inventory. One of the renamed items will drop, because the items actually didn't properly stack: Instead the second renamed item on the cursor switched the position with the clicked item in the inventory.
• After dragging the second renamed item around inside the inventory will, the items will properly stack.

I was able to reproduce this in both creative and survival mode. The server is up-to-date and there are no plugins.

Vanilla minecraft doesn't seem to be affected by this.