Can you please stop with the edition warring? Is that really too much to ask?
No, 5e does not talk about success at a cost, "just like 4e". It really doesn't. 5e presents skills as binary pass/fail. There's a bit of advice buried in the DMG about the notion of degrees of success or failure, but, there's nothing about the notion of success at a cost. There's barely any advice on the notion of group checks.
But, then, why would you want any sort of success at a cost? That's just rollplay, right? After all, you can only have success at a cost if you're rolling checks. ANd, you have REPEATEDLY claimed that that's rollplay and not something you want to do. So, which is it? Freeform roleplay or roll play?
You were the one saying that a skill challenge would resolve the issue. I'm just stating my opinion that a systematic approach would not solve the issue. Either the DM is trying to give the player the hint that what they are doing will eventually fail, or their just a bad DM. As I've stated repeatedly I use complex skill challenges all the time that are similar to skill challenges, I just don't use a scorecard of successes and failures.
As far as what the 5E DMG, I was talking about Success at a Cost in Chapter 8.