One of the lesser issues in 5e (and also earlier editions) is that if you have two characters, one with proficiency in a skill but a low attribute and the other with no proficiency but with a high attribute, the second of these has a better chance to succeed at relevant skill checks than the one who has invested actual resources in getting 'good' at that thing.
For example: Fred the Fighter has a Cha of 10 and is trained in Intimidate. His skill modifier is therefore +2; Bob the Bard has a Cha of 18 but no proficiency; his skill modifier is +4.
Of course, it's valid to declare this to be a feature rather than a bug. It's also valid to either have a bunch of things that either a proficient character can succeed at automatically or that, conversely, an untrained character automatically fails.
But here's a proposed alternative: if you do not have training in a skill (tool, whatever), you get to apply your attribute bonus up to the maximum of your proficiency bonus.
In which case, both Fred and Bob would end up with the same skill total of +2.
This has the effect of allowing an untrained character to be as good as, but never better than, his trained companion. And, since most characters will also have at least a small bonus in their relevant attribute they'll usually be just a little behind.
Plus, it has the advantage of being a nice, simple rule that cleans up a minor annoyance.
(Of course, it does also raise a question: does this also apply to attack rolls, saving throws, and armour use, all of which have a proficiency mechanic? Though, in this case, I think I'm going to say "no".)