No, not wanted in the criminal sense.
What I mean is, should you consider how eager a school is to have you in your decision? As a hypothetical, say I have two schools, A and B. Suppose they are roughly similar on fit, funding, ranking, ect. (But not so similar that you couldn't decide on those factors alone.) In other words, a close toss up. School A gets back to you less than a week after the interview, with note from the PD really trying to sell the school to you. School B sends you a short note several weeks after interviewing that you've been placed on the waitlist; a few weeks after that, offers you a spot (without so much selling of the program).
How much (if any) weight should be placed on the fact that one school seemed more interested in recruiting you?