Take an infinite set of distinct primes and a (edit: or 2 , etc.) residue class for every prime. For exammple you can take all the primes bigger than some prime or the primes of a specific form (i.e. of some arithmetic progression or whatelse)

The question is under what circumstances every natural number is of some of this residue classes?

Of course there are conditions that the answer is easy(for example if you take as residue class for a prime on the set to be $1\bmod{p_1}$, $2\bmod{p_2}$), but do we have any well known general results on this direction? (On infinite sets of residue classes etc.)

Note:if you think that the question is not very specific please ask me to be more.

EDIT: to be more precise i will give an example : Take all the primes from a point and after, split the natural numbers on intervals of a given length, say A, and for each prime is not allowed to "hit" a number that is in lower interval than itself, (i.e take all the primes greater or equal than 7 and $A=100$)