Some engines may have lower oil pressure at full-temp than ideal probably that's your first concern. In colder environments; e.g. during winter up north in US/Canada, then an app that generally runs 5w-30 may be just fine with 0w-20 if you rarely see temps above freezing and get those single digit and minus readings a lot. However, this is mainly an improved cold flow and for short tripping. If you still see severe duty in the winter in terms of workload on the engine; even though colder ambient, you will still run into the concern over dropping the grade low to the point its HTHS numbers may be somewhat concerning and you'd probably first see oil pressure readings dip lower than you normal see at full temp with 5w30, etc.
That's the gist anyway. The specifics depend on your app, age of engine, how much wear is already present, etc. A tired oil pump may not like going thinner if the OE grade is showing minimums on the oil pressure. For example, this engine approaching 300k and being a 1992 I don't really want to risk the oil pressure dropping too much by going to a 0w-20. Is it possible? Sure. Do I want to risk it? No. Maybe I am a bit too conservative, but it is my only vehicle, so...someone else first?

Honestly, I could get away with it BUT as I said if my oil pump is tired that'd probably be the first thing to show and I don't REALLY want to risk oil pressure dropping and having any more wear to the crankshaft/bearings/rods at this age than need be. At some point, you want to give yourself more head room for protection. In newer vehicles that can handle whatever full-temp discrepancy 'may' be present I don't think you'd run into any problems long-term with regular service. Again, YMMV per app and usage.