For the most part no, but it depends on the job. If your goal is to go to a very academic setting - like the NIMH - or if your job ambition is to become a highly published department chair or dean or something - it would probably help to know people in academic circles. Even then it may not be critical for "entry level" positions in those settings.
For the 98% of jobs outside that, don't think it matters much as long as you're a good fit/hard worker/competent doc. As a peer, when I'm interviewing someone to join my practice/academic department I could care less where they come from, I just want to know 2 things - (1) If they're competent enough I'd feel comfortable referring my wife/kid/mom to them, and (2) if my black cloud service has gone nuclear at 7PM on a Friday, and they're done for the day, are they going to help me out with a good attitude and let me buy them a thank you beer afterwards, or are they going to leave me in the weeds.
For the most part department chairs/program directors/practice heads (I think) have the same kind of thoughts when hiring. If you're filling high level spots in a name brand heavy research/academic institution, it's all about funding/grants/publishing. Outside of that, bosses want to hire people that aren't going to give them problems. If you can demonstrate you're a good fit, good team player, good attitude, etc... that goes a lot further than if you trained at East Tierra Del Fuego Community Mental Health Center vs. Mass General.
--Random thought 1--
What matters the most (IMHO) is location - if you want to work in a certain area, you're much more likely to get a job there if you train/do fellowship there - you'll meet more people and get wind of openings much sooner.
--Random thought 2--
Your "last" thing carries the most weight. For residency applicants your med school carries more weight than your college; for fellowship applicants your residency matters more than medical school; for your first jobby-job your fellowship matters more than your residency; for subsequent jobby-jobs your last jobby-job means more than anything else.
--Random thought 3--
If you really want to go private practice/group practice, you may actually be at a disadvantage coming from big academics. From my (limited) experience, if I'm hiring a doc to join my purely private practice make money group, and I've got an applicant from Mass General vs. one from Hollywood Upstairs Medical College, all things being equal I'm taking Dr. Nick, because odds are he worked with a lot of community docs in training, and has some idea of how the basics of that sort of practice work.
--Random thought 4--
Given all the stuff above, experience *does* matter, and fellowships *do* matter. For the most part, all things being equal, I'll fill my hospital's vacant C/L spot with someone with a fellowship over someone without. The "official" salary of spots may be the same, but if you and I are both taking the same job in the same setting, and I've got 10 years of successful work experience on you (published articles/RVU's/$$$ practice/whatever) I can almost promise (from everywhere I've been) I'm either getting some kind of money or "admin time" or whatever over you.