Yeah, I did tread all over it, because it's wrong. The quote presents a single story, and within that story you're right and Bruce is shown to be right. We can push ourselves.

There's no mention of the actual physical status of the narrator, or how he felt afterwards. He could have actually been on the verge of a heart attack, and if things had gone a little differently he might have actually had one.

Besides that, yes, the human body is capable of pushing it's normal limits - always at a price. A lot of people are familiar with the idea of adrenaline induced strength, which makes people suddenly capable of superhuman feats. Did you know that it also severely damages the muscles that were used? Hell, a more mundane example - if you try to do too much when you're working out, you hurt yourself.

If your limitations are only self-imposed, then it is valuable to push them, but from there comes the wisdom of knowing where your limits are. This quote has nothing to do with that, and ignores it like a child who wants to believe in a simpler world where people can do anything they want with a little grit.

This quote is about a man telling another man who may well have been put in mortal danger by that level of exertion to die. Bruce Lee is being a prick. He's in good shape to do 5 miles. And yet he stops at 5, and belittles someone else for struggling. Maybe the narrator was fine and just complaining, sure, those details are ambiguous. Maybe it was appropriate for that situation. If you take the sentiment to heart, though, then there are a thousand situations where you will put yourself in danger for the sake of denying that limitations exist.

TLDR; Pushing yourself to achieve your best potential and ignoring limits are too very different things.