Just like that surprise "extra menu" that pops up after loading 12 games onto the Switch, users have begun to discover a new screenshot limit on the system. With folks taking literally hundreds of screens from Zelda: Breath of the Wild it was bound to happen, but reddit user megamilkybee has confirmed what that limit actually is.
According to a warning that pops up in the album app, the system memory can only hold 1,000 screens, lest you move or delete them. If you pop in an optional microSD card you can hold up to 10,000. Provided that you're not completely attached to having the screens on a card you can (turn off your system), pop the card out, copy them all onto your PC, then wipe the screens folder clean.
That should be more than sufficient going forward, just keep in mind that your options will be limited if you aren't getting a microSD card anytime soon (maybe you're waiting for larger capacity cards to drop?).
I hit my screenshot limit [Reddit]