I try my best to use Pokemon I rarely use. Going off memory, I believe I started to incorporate at least 3-4 underused Pokemon per team on my initial playthroughs back in HGSS.
In my original HG playthrough, I used Meganium, Pidgeot, Ampharos, Starmie, Nidoqueen and Scizor. Outside of Starmie and Scizor, the other members faced their struggles here and there, even with as much EV-training as I could squeeze in. Starmie/Scizor just more or less swept whatever they faced. Meganium had coverage issues, my Pidgeot was too squishy, Ampharos was too slow, and Nidoqueen didn't pack enough punch.
Sometimes I try for themes, as well. For instance, in my current replay through X, I only use Psychic types. Espeon, Alakazam, Gardevoir, Gallade, Starmie (for Surf), and Mew (for Fly). I had the Mew as the "starter," transferred over from HG. But I admit I consciously avoided unaesthetic Pokemon like Mr. Mime, Jynx, Hypno, Blobuffett.
Current replay through Y, I run Ninetales, Altaria (no mega, obviously) and Kabutops. Those three are probably the weakest links on the team (the others are Greninja, Roserade and Scizor). Seriously, I love my Psyshock Ninetales, but ugh, your stats blow.
The all-Psychic team I'm using for X is actually the first time I've ever used a legendary on my team since my original playthrough of Blue version (where I carried all three legendary birds on the first run). Usually I don't, but since I was going with a theme, I needed a Psychic type that could learn Fly and actually use it properly (i.e. didn't use Sp.Attackers Xatu, Swoobat, Sigilyph). Other choice would have been Lugia/Latios/Latias, but I was using a chick for my character and felt a tiny Adamant Mew made more sense as a companion. Yes, I put a lot of thought into it.
TL;DR I wouldn't say never for any Pokemon, but some are more unlikely than others to find a way onto the team. Maybe one of these days I'll choose to run an all unevolved team (stage 1 out of 3).