Which problem? Yes, there's always disk contention. But if you don't have swap enabled then you won't have pathological swap activity dominating the disk.

There are several bad things that happen when you run out of memory. One of them is pathological swapping. Another is loss of disk caches. Another is (eventual) oom. The one that has the most detrimental effect, and typically results in the system having to be reset, is swap activity saturating disk io. Even when you've got fast SSDs, because they're still orders of magnitude slower than memory.