Alternate Universe Answer:
QB: Rich Gannon, Fran Tarkenton, Andrew Walter, Tim Tebow, Matt Flynn
WR: Torry Holt, Reggie Wayne, Gary Collins, Nate Washington, Andre Holmes, Michael Jenkins, Brandon Gibson, Anthony Gonzalez
RB: Duane Thomas, Rocky Bleier, Tommy Mason, Joseph Addai, Donald Brown
FB/TE: Bear Pascoe
TE: Marcus Pollard, Alex Smith, Lee Smith, Jake Ballard
OL: LT Bob Vogel, T/G Dan LaRose, C/G Ted Larsen, G/T Robert Gallery, RT Corey Hilliard, g Ken Byers
DE: Will Smith, Fred Dryer, Jonathan Fanene, Jeremy Mincey
DT: Phil Olsen, Christian Peter, Terrance Knighton
LB: Lee Roy Jordan, Rip Hawkins, James Anderson, Victor Hobson, Justin Rogers
CB: Bernie McRae, Will Allen, Eric Warfield, EJ Biggers
S: John Lynch, Adrian Wilson, Tank Williams
K: Robbie Gould, Martin Gramatica
P: Gary Collins
KR: Bill Triplett
All of the above players were either once part of the Patriots' organization, at one time traded to the Patriots, or drafted by the Pats - but never appeared in any games (other than preseason) with the Patriots.
And yes, I know there are too many WR and QB, and not enough OL.