It's up to the DM to provide resources to the players, and it's up to the players to best choose how to disburse those resources. If they choose poor disbursement -- like giving four attunement items to one player -- it's not up to you to change to rules to make it better.
I'd only consider increasing attunement if multiple players had hit the limit, and I'd probably resolve the issue by eliminating attunement requirements from some items (e.g.,
ring of feather falling,
ring of jumping, et al) rather than increasing the number of items players can attune. I would probably say that the items simply must be worn during a short rest to begin functioning for that character -- virtual attunement -- instead of requiring true attunement.
See, there's two reasons for an item to be an attuned item:
- Items that the designers think make a character powerful (typically just in combat, but possibly out of combat).
- Items that the designers don't want passed around the party.
Items in the first group should always be limited to 3 for attunement. The game isn't built around large amounts of +X to game stats abilities and layering additional defensive and offensive abilities, and lots of magic items do that.
Items in the latter group, however, just make it more difficult for players to pass around the party. You don't want the party to find a
ring of jumping or
gloves of climbing and swimming and let the players overcome an obstacle, throw the item to the next player, rinse, wash, repeat. Now they've turned one item into five items. That's a lot more powerful than the item is really intended to be. Now, they
can do that if they feel they need to, but it will be time consuming. The classic items that have always been restricted like this are
ring of sustenance and
ring of regeneration. The latter probably fits in the first category, too, however. (An item can be in both categories, of course.)
I would say that a magical bow, a
cloak of displacement, a magical short sword, and
bracers of archery all fall into the first category.
If I were playing in a very high magic campaign and decided it was necessary, I'd probably change attunement limit to be equal to proficiency bonus. That's better when it's relevant so it does what's needed, and also easy to remember.