I can suggest one more thing.
Sometimes RL apps do not properly identify normal/bump maps when importing.
You have to reload them manually and make a proper selection (normal or bump).

Do the following:
Right mouse click on bump map, then select Copy.
Right mouse click again and select Paste.
At the prompt do a proper selection (Bump in you case) and see if that made a difference.

Looks like you are correct. I don't know what I saw earlier...Importing a 2048 x 2048 texture and the exporting the texture does retain its size.

"We often compare ourselves to the U.S. and often they come out the best, but they only have the right to bear arms while we have the right to bare breasts"
Bowser and Blue, Busting The Breast