No if say a piano player gives a note (provided its in your range) and you can't get in tune with it, you just are jumping until you think your right, and you never make it. You'll need relative pitch if you were, say, sight singing music, and you get a starting note.

Now I agree that vocal instructors don't teach pitch, but there is still hope if you don't (which is where I differ from iiholly).

I'll start with a story of mine. A few months back I was the student director of a junior version of the musical Into the Woods (It was HELL...). Now there was this one kid who was completely tone deaf, yet he had been cast with a half of a song for a solo (by the teacher director, who did the castiong, then handed it to me and took care of stage crew 'n' stuff). So we had to figure this out. Being the most musical person in the performance (as in, I was the only person who could tell if something was sharp or flat) and the teacher director never helping us, I got elected by the kid's girlfriend (also in the play) to help him out. Lucky me :D ! So what I did was take the sheet music (I don't play piano, but I know the notes) and took him over to the piano. First I found out the key of the song. I went to the first note of the scale, and had him sing it. Way off, as was expected. So I told him to stretch the note until he had it on (I told him flat, go higher, sharp, lower etc...) Eventually he got it. Next note in the scale, same thing, we did that for probably a half hour to an hour (so this is like half the rehearsal already), but as we did it more, he started to become quicker on getting the notes right. It was kind of like an ear training that used singing as well. Eventually, he became much better at the song. He also built up confidence for other stuff.

Where was I going with this? If you know you are completely tone deaf, and have a friend with a good ear, who's not afraid to say you are nowhere near the note, you can get better at pitch recognition, enough to be able to sing halfway decent. Of course, you probably will still be a couple steps behind, because the notes won't register as fast as people who are not that tone deaf.

Hope that helps you!