So I was studying quantum mechanics and I came upon this table that shows the probability (T) of a given particle tunneling through a potential barrier. And the last value $10^{-628}$ made me think about this question: Just like there is a "lower bound" on time and distance where physics stops making sense, is there also a "physical" lower bound on probability where we can affirm that an event is impossible?
No -- if the probability for an event is merely small, but non-zero, then the event can occur, and given enough time, will occur.
But, there is some common sense we can apply. One way to interpret a probability is in terms of how surprised you might be about the result. If you do an experiment, you wouldn't be surprised at all to see an event that had a $1/3$ probability of occur; you may be somewhat surprised to see an event that had a $1/20$ chance of occur; you may start thinking that there is something you hadn't expected going on if you saw an event with a $1/1000$ chance of occur. Those thresholds roughly correspond to $1$, $2$, and $3$ sigma. This tells you how likely an observed event is to occur given a null hypothesis. To be able to claim a discovery of a new particle in high-energy physics, one of the criteria is that the probability that the default model, without this new particle, is only able to produce the observed data with a probability of about $10^{-6}$ or less, or $5$ sigma.
If you saw an event which quantum mechanics predicted had a probability of $10^{-628}$ to occur, you couldn't conclude that the laws of physics had completely broken down, but you may want to carefully double check what you saw, and also think about whether something is wrong with the calculation of the probability. Maybe there is some other effect that can look like a proton tunneling that is more common (wouldn't take much to be more common than $10^{-628}$), maybe you had a loose cable in your apparatus and you didn't see what you thought, maybe quantum mechanics is wrong and the true probability of the event is higher. Or, maybe you really did just get unspeakably lucky (but you had better work very hard to rule out other explanations before you simply believe that).
Another practical consideration is that observing an event with a probability of $10^{-628}$ would be essentially impossible to replicate, which would make it difficult to believe the result. As an amusing anecdote, Blas Cabera claimed to have detected magnetic monopoles one Valentine's Day in the 1980s (https://www.nature.com/articles/429010a). As far as I know there is no proof that he didn't see a magnetic monopole, but the probability of seeing one (if they exist) is so small that no one has been able to replicate the discovery, and so we still don't know whether magnetic monopoles exist or not.
Yes, indeed, one can safely neglect any event that is unlikely to happen within the observation time of our experiment. What constitutes an observation time depends on the situation:
Examples
