Well I'm probably not the best person to ask as I shall be well into my 60's by the time I make it out of training grades... The short answer is that (at least where I am in the UK) there is no cut off as long as you retain the necessary neural plasticity to learn as rapidly and as retentively as a 20 to 30 year old.
I suspect the USA is exactly the same. Here we have an entrance exam for mature entrants, which is kind of the equivalent of running an intellectual triple marathon. Less than 10% of those who sit it pass it... and less than 30% of those who pass it, get a place after interviews. I managed both despite my age.
To be honest most people of my age probably could NOT cope - but for some reason I seem to have managed to defy the ageing process, must be the timelord genes I guess ;-)
That said there is actually a lot to be said for being older. Youthful energy and swiftness of thought is great and will get you a long way... however there are some situations where the extra maturity and life experience serves me well and gives me a distinct edge over my young rivals.
For example I know how to talk to patients who are stressed and frightened. I know how to read older people, I can often pull off the appearance of greater authority than perhaps my younger colleagues and as a result a lot of patients find it easier to trust me...
I've seen a few things in life, so I don't get so affected by some of the, frankly awful, things that I witness. So its roundabouts and swings. I wouldn't advise leaving it until you are in your 50's like I have - but certainly there is nothing wrong with doing it in your 30's.