Yes, in real life you can absolutely wear, and almost certainly conceal, a chain shirt under normal clothes. In D&D it's - as with all things, up to your DM. However, consider: -
For starters, the Player's Handbook says:-
a chain shirt is worn between layers of clothing
(emphasis added by me)
A chain shirt, or "byrnie", is a t-shirt-length item. It's not a long-sleeved garment, and it's supposed to be quite close-fitting to the body.
Let's address the issue of bulky padding. While it is undoubtedly more comfortable (in terms of getting hit, not say, heat exhaustion) and effective to wear armour over padding, the arming jack and later arming doublet (gambesons are generally a separate, stand-alone item of bulky padded armour) appear about a thousand years after the invention of mail armour. Celts, Romans (well, a "sub-armalis", a thicker than usual tunic), Angles, Saxons, and even Vikings until about the Norman invasion, wore a simple tunic under their mail.
When I do Dark Ages LARP and re-enactment, wearing real mail armour (riveted, steel, mail) I wear it over an old tunic - you don't need to wear significant padding under it to stop sword cuts.
Lindybeige does a video on mail, where he wears mail under his sweater, and only reveals he's wearing a mail byrnie/mail shirt halfway through the video. I'm linking it here https://youtu.be/RssIl2v0C1k
As with a lot of D&D questions, while the text hints at the possibility of it being "muffled by outer layers", it's down to the DM to make the final call. If I was DMing, I'd allow a chain shirt to be worn concealed, but allow NPCs to have a chance of noticing it if they were really looking. As for full mail ("chainmail"), no, since it's a lot longer and harder to conceal, and in 5e includes that padded arming jack or "gambeson" [sic], and doesn't explicitly mention other clothing covering it.