I notice that it says that "Untitled," which I assume is my backup disk, has 925 GB used.
I do not know why it says "Untitled" unless that is the name that you assigned to the drive.
Is it possible that it doesn't have room for another backup?
Yes. If the disk is 1 TB and you have used 925 GB, it is very close to full, and Time Machine requires a lot of free space in addition to the space for the backup to occur. But, if that is the issue, you will usually see a message to the effect that "There is not enough room for the backup to occur".
If it's full, wouldn't it just start deleting old files to copy new ones?
That's the theory, which does not work all that well in practice. IF.....a backup is a bit larger than a normal small incremental backup......which could occur after an update, new images have been added, etc.......Time Machine cannot delete enough old files to make room for the new backup.
I believe that after some discussion, that Apple Support will ask you to erase the backup drive and start over again with a new backup. Hope I am wrong.
Let us know how things turn out if you can. Good Luck!