I don't believe late 1st round picks are cut all the time. Sure they often don't get a second contract, but cut before the original 3 years is actually fairly rare.
What are you talking about, it happens all the time Fab Melo, JuJuan Johnson, JR Giddens, Gerald Green. Oh wait you're talking about non-Celtics players
But seriously though, I think it happens enough. You say 3 years and are specifically talking about being cut, but I would use 4 years, as 1st round rookie contracts are 2 years guaranteed plus 2 one year team options before the player hits restricted free agency. So going back 10 years, how many 1st round rookies didn't make it past 3-4 years in the league?
2006: 6 guys made it 3 years, 6 guys made it only 4 years
2007: 4 guys made it 3 years or less,5 guys made it only 4 years, 1 guy has never played
2008: 2 guys made it 3 years or less, 1 guy made it only 4 years
2009: 3 guys made it 3 years or less, 2 guys made it only 4 years
2010: 4 guys made it 3 years or less, 3 guys made it only 4 years
2011: 5 guys made it 3 years or less, 0 guys made it only 4 years*
2012: 5 guys made it 3 years or less, 24 guys haven't made it past 4 years! Oh wait I have to stop counting at this point.
*excluding foreigners who came over later like Jonas Valanciunas, Nikola Mirotic, etc.
It seems to me, depending on the year, anywhere from 10% to 40% of first rounders don't make it past 4 years in the league (though I'd say the average is probably closer to 15-20%). And then we have a variety of reasons of why that happens.
In some cases you have injuries (Jonny Flynn, Greg Oden) or other issues (Royce White, Javaris Crittenton) keeping a player from playing, but on the flip side you have a lot of 10 day contracts keeping a player in the league to even get up to or past 3-4 years at all (DJ White, Terrence Williams, Elliot Williams, Jordan Hamilton, etc.). Some times a players contract is just allowed to run out and/or the team option isn't picked up (Joe Alexander, Patrick O'Bryant), other times he's shipped around the league as trade filler, and some just happen to have a dad whose a GM.
Again, I know you specifically said players being cut on rookie contracts, so already we may be talking about apples and oranges here. I'm just saying I would broaden that criteria a little, and a decent amount of teams would have "wasted" a first rounder on a player that they ended up getting little-to-no value from.