Not in the slightest! The answer is not even known for quadratic imaginary number fields. In fact, the only known way to show that the Hilbert class field tower of a number field is infinite is to invoke one of a variety of different forms of Golod-Shafarevich, and I don't think it's even seriously conjectured (more like "wondered") that every infinite Hilbert class field tower arises by applying Golod-Shafarevich to some step in the tower (or to some cleverly chosen subfield).
Incidentally, the "sufficiently many primes ramified" business is a bit of a red herring, in my opinion. The real condition is that the $p$-rank of the class group is large for some prime $p$. When $K$ is cyclic of degree $p$, it is only the fact that genus theory relates the $p$-rank of the class group to the number of ramified primes that brings ramified primes into the picture. (For example, the standard Golod-Sharevich examples come from showing the 2-class field tower is infinite by using Gauss' result that many primes ramifying in a quadratic extension imply a large 2-rank). For non-cyclic extensions, the link is more tenuous, and it becomes much more natural to talk strictly in terms of the class group.
