According to draft value charts, 6th round picks don't carry much value. Certainly more than the lousy ONE draft value point that is the majority of the 7th round...but not substantial draft value points. Even in your scenario, you'd have to be using a pretty high 4th round pick to be getting yourself above the bottom of the 3rd round. Of course, the reality is, draft day trades are not governed by some sites' draft value chart. So, if you can find a sucker, er, I mean a trade partner, well, you're golden. And that does happen at every draft. Just not very often.
But I do see veteran players being traded for 6th round draft picks. That's why I say we'll probably do something like that with one or more of those low value draft picks.