It will depend on the restaurant and the crowd levels. Since we have AP and go quite often we usually don't plan ahead and get an ADR before we go. I have called Disney Dining while at the parks for the same day and gotten in at WTC several times with no problem. You have to make ADRs at least one hour before you want to arrive.
One time we were in line at GRR and I called Disney Dining from my cell phone. We were able to get an ADR in one hour and after we got off GRR went straight to WTC. We checked in and only waited 2-3 min. for our pager to go off. As we were being escorted to our table I overhead one guest ask "why didn't they have to wait?" She was told we had made reservations. I loved it!
I've also called from home the morning of and was able to get a table for dinner at WTC, in June, just in time to watch the EP after we ate (it was my birthday

).
In May last year I called the day before and got ADRs at New Orleans cafe the day CELEBRATE premiered. Had to take 4:30 but it was still a good time. The weather was nice and the food was good.
Conversely when we go to WDW (with free
DDP) I will make at least one ADR for each day. We do end up changing some of them but it's nice to have at least one option set. It's not as important at DLR but if there are places you really want to go, (Ariel's, Goofy's, etc) go ahead and make the call. Couldn't hurt.
HTH.
(Note-I use ADR (advanced dining reservation) for both WDW and DLR. I think it's actually called PS (Priority Seating) at DL. Same thing, different terminology)