In the U.S. all of these would be correct: reach capacity reach their capacity reach the limits of their capacity reach their capacity limit/limits (is grammatically correct, but not usually phrased that way)

Kayli, I agree with your explanation of "reached." However, in the example he gave, you would not say, "What are we to do should the hospitals REACHED their capacity?" You are talking about the future, so you would not use the past tense. It would be "reach."

The former...

Here's my gut feel, as I'm not sure there's a huge difference between the two. I'd use "hospitals reach capacity," or "hospitals reach THEIR capacity limitS." Or, if you intend to say it with a slightly different meaning, you might say, "hospitals reach their limited capacity" (which puts more emphasis on the fact that there's only a limited amount to space available). Hope this helps?

It is reached capacity. Because the act of getting to full capacity in a place has already happened thus it is in the past. example: This hospital has reached its capacity for the number of patients it can treat. Most hospitals will have a capacity limit which is the maximum amount of people that a place can hold. We also often say "It has Reached maxium capcity" to show the upper limit.