If you have more than one connection within a company, should you use them both or just one? (applying, interview)

[NHHiker1]

Let's say you are applying for a new job at a company. You also know a couple of people within that company who would be willing to put your name out there and vouch for you.

Do you just pick one? The most well-connected, influential one, for example? Or if you know multiple people, should you try to get them all to go to bat for you? Would this make you look better or would it come across as annoying if you've obviously got multiple people trying to get you in?

[jaypee]

IME, it's better to have as many recommendations as possible.
The last time I referred someone, the recruiter said "You're the third person to mention this candidate. We need to get him in here for an interview asap."