Yes!! But if you counter make sure you have a backup too. Also, check averages for the role before  negotiating

Partnership manager

Is it an HR trick to avoid salary negotiation by saying that they're already offering you out of budget for the band, and that none of the others on the team

Is it an HR trick to avoid salary negotiation by saying that they're already offering you out of budget for the band, and that none of the others on the team earn as much? "Please keep it to yourself" 
Had this happen to me twice and can't help but think it's just a line

Negotiating salary