Not as a guaranteed benefit. If it were to happen, it would be completely at the hotel’s discretion. I will say, though, that I’ve gotten numerous upgrades to premium suites even when booking standard rooms, so it’s certainly a possibility. Standouts off the top of my head were PH Milano, PH Washington, and HR Churchill in London.

You could also book via Prive to get an automatic upgrade to a suite one step above what you booked, though with some restrictions - you’re likely not going to get upgraded into a Presidential suite just because you book the suite one step below it.