If you're trying to do an exact calculation, then no, you should not ignore any of the possibilities. Of course, if you just want an approximate result, ignoring low-probability events may be justified, as long as the effect of those events on what you're trying to calculate is not disproportionately large.
For example, say you're playing a game where you toss a fair coin repeatedly. Every time the coin comes up heads, you win \$1; the first time it comes up tails, the game ends. Then your expected winnings from the game are $\frac{\$1}2 + \frac{\$1}4 + \frac{\$1}8 + \frac{\$1}{16} + \dotsb = \sum_{k=1}^\infty \frac{\$1}{2^k} = \$1.$
If you truncate this series after $n$ terms, the sum will not be exactly \$1. But it won't be off by more than $\$1/2^n, so the difference becomes negligible (less that one cent) after only 7 terms.
However, let's say you're playing a different game, where you win \0.01 on the first heads, \$0.02 on the second, \$0.04 on the third, \$0.08 on the fourth and so on. You might think that this game doesn't pay as well as the previous one, since the rewards are in cents instead of dollars, and since the game is very likely to end before they grow much larger. But if you calculate the expected winnings from playing this game, they are $\frac{\$0.01}{2} + \frac{\$0.02}{4} + \frac{\$0.04}{8} + \dotsb = \sum_{k=1}^\infty \frac{2^{k-1}}{2^k}\cdot\$0.01 = \sum_{k=1}^\infty \frac{\$0.01}{2} = \infty.$
This despite the fact that, if you only play up to $n$ rounds of this game, your expected winnings are only $n/2$ cents, and that the odds of your game lasting more than, say, 20 rounds are literally less than one in a million! However, if you do manage to toss 20 heads in a row, you'll already have won over \$10,000, and the prizes just keep going up from there.
The example I used above is known as the St. Petersburg paradox. You can find a lot of interesting discussion about the implications of this paradox, but the message to take home from this example is that you shouldn't neglect even extremely unlikely events if the effects of those events, if and when they do happen, is equally extreme.