I recently read about the remarkable 15 and 290 theorems, the second of which states that any positive definite integral quadratic form represents all positive integers iff it represents all positive integers $\leq 290$. (The 15 theorem says that, if such a form can be represented by a symmetric integer square matrix (arbitrary integer quadratic forms can correspond to half-integer symmetric matrices via (half) the Hessian), it suffices to consider the positive integers $\leq 15$.)
Is there any reason why we should expect this theorem to be true? I doubt there's a concrete reason to expect $15$ or $290$ in particular to be the "cutoff" points - for example, $33$ is the largest integer that is not expressible as the sum of five positive squares, but as far as I know there isn't a particular "reason" why (one can prove that all large enough $n$ can be and then can manually check the rest) - so I guess I'm looking for a reason why such a cutoff should exist:
Why should there be a positive integer $N$ so that a quadratic form reaching all positive integers $1\leq n\leq N$ implies its universality?