> 2 billion is a very large number that was probably not envisioned as reachable in the near future

I disagree. Simple napkin calculation: 100 million players playing 40 games each per year (about 1 per week) over 5 years = 10 billion unique games.

As others pointed out it was likely not a miscalculation, just a lack of calculation. The bug occurred only in the client and the decision to use a smaller data type was likely not a conscious one.

In any case, I wouldn't hold it against an individual programmer. But arguably this sort of bug indicates your development process has flaws (not enough testing, code reviews, etc).