According to David Aldridge of TNT the Nuggets attempted to move up to the #3 overall pick but decided that the price was too steep.
Denver tried to move up to three, but like everyone else, for now, felt Boston wanted too much for the pick(s).— David Aldridge (@daldridgetnt) June 23, 2016
Most likely a deal with the Boston Celtics would be centered around Danilo Gallinari and probably also involved the #7 pick. While Denver certainly seems to be interested in moving up, they likely believe they still have a chance of getting their guy at the seventh spot. We'll probably never know what the offer actually was but if Tim Connelly decided against it he likely was making the right decision.
Still, the Nuggets are keen on getting themselves a shooter in the draft and the best two Buddy Hield and Jamal Murray, could be gone by the time the Nuggets get to make their selection so there is definitely interest on the Nuggets part to move up and land one of those guys. In the end though, it needs to be a smart trade, and not just a trade for trade purposes.