The driver have thermal thermal protection, that switches the current off until the cool down and then switches them back on. This effect is similar is some ways to skipping steps. It does not immediately damage the board, however your print will have layer shifts and other nasties so its not a useful way to run the drivers. Also constantly running electronics right at its thermal limits will reduce its life over time.

Have a look at your motor data sheet, you may find that the 2.0A is @85C which is hot to run the motors at. At that temperature if you have your motors on printed mounts you may start to see softening of the plastic and the mounts will creep over time.

That is why most people run their motors well under rated current.