On 10 night from 8 stays, I have seen a suite twice. Once was last night at the PH DC where, without asking, I was booking into an Executive Suite when using 1-800-CHECK-IN.
However, I have usually received a very large room. My room at the PH Philly was bigger than many suites.
My only problem so far has been at the Hyatt Crystal City. I stayed there on a $69 Hyatt.com rate and my status was not acknowledged until I asked about breakfast vouchers. The desk agent asked if my rate included breakfast. I told her that I was a diamond and she took a minute to look at the screen and then gave me two vouchers. She did not appear to try to find me a larger room though. The one I did receive was on the "wrong" side of the hotel and was pretty small. The next day someone knocked on my door a little after noon and asked why I was not out of the room yet. I told him that I had asked for, and received, a 2 pm check-out. He apologized and let me be, but it still was an indication (along with check-in the night before) that this hotel needs to train their staff a little better.
That said, all other stays so far this year have been very nice. I do not believe that suite upgrades are a published part of the diamond benefits, so I do not ask for one. However, I think one would generally have good luck if they ask the 1-800-CHECK-IN agent for a suite.