After changing my user name to D B Lim, it seems whenever someone does @D or @D B Lim, I don't get a notification. Is this some kind of glitch? Should I change my username to one that is just one word?
You need to remove the spaces to get the notification working if the first part of your username is only one letter, so
@dblim should work.
See How do comment @replies work? for much more details about the comment notifications.
Click the "help" link underneath the Add Comment button:
The post author will always be notified of your comment. To also notify a previous commenter, mention their user name: @peter or @PeterSmith will both work. Learn more…
The implication is that the user's name is "Peter Smith" so the lack of space is indicated there.
Additionally the Learn More link to https://math.meta.stackexchange.com/editing-help#comment-formatting is HIGHLY recommended.