I think a lot depends on how busy the actual ride is. We were about 5 minutes early for our fast pass at BTM a few weeks ago, and we were kept outside until the exact time. But, I think it was because BTM had broken down earlier, and the fp queue was absolutely huge with people coming up from earlier times they'd missed.
I haven't tried after the FP time though, but have seen people turned away. Again this is usually at BTM where they check your time thoroughly.
On the plus side - when using the DLH hotel fastpass, our dates weren't even looked at, so we managed to use some from days we'd missed before.