Infinity is not something that easily makes sense conceptually. Consider the Hilbert Hotel.
http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel
Basically you consider a Hotel with an infinite but countable number of rooms. That is, it has a Room #1, Room #2, Room #3, and so on, such that there is a room corresponding to every positive integer. Every room of the hotel is occupied.
The paradox is that even though every room in the hotel is occupied, you can still accommodate a new guest. If a new guest comes, simply tell everyone to move one room down. The new guest can then take Room #1.
The hotel can also hold 2 new guests -- tell everyone to move down 2 rooms. You can extend this to say that the hotel can hold any finite number of guests by having the current guests move down a certain number of rooms.
The hotel can also hold INFINITE additional guests. If we have an infinite (but countable) number of guests arrive, have the current guests occupy only the even numbered rooms. There are infinitely many even numbered rooms, so all can be accounted for. And now we have all of the odd numbered rooms free, and there are infinitely many of those so we can assign our new guests to those rooms.
So yes, it is normal for it to be confusing to get your head around the idea.
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No amount of planning will ever replace dumb luck.