I don't believe the answer to this question is known. There are various things one can say that are related. For example, there are known non-zero elements of known order from the image of the J homomorphism in all dimensions congruent to 3 mod 4.

So none of those groups is zero, and if you like, you can then say that there can't be more than three consecutive zero groups.

There are other conclusions like this that one can draw, but I don't know how to show that all dimensions congruent to k mod 4 are non-zero for any k other than 3.

I don't believe the answer to this question is known. There are various things one can say that are related. For example, there are known non-zero elements of known order from the image of the J homomorphism in all dimensions congruent to 3 mod 4 (by which I mean $\pi_{2+n}(S^2)$ with n congruent to 3 mod 4).

So none of those groups is zero, and if you like, you can then say that there can't be more than three consecutive zero groups.

There are other conclusions like this that one can draw, but I don't know how to show that all dimensions congruent to k mod 4 are non-zero for any k other than 3.