I've done this before, both with same day "connections" and next day "connections" (on separate bookings).

My domestic flight to LAX on AA was delayed with unknown ETD/ETA, and I was very stressed out and booked a separate one-way on Southwest to LAX at the last moment. My checked luggage took forever to arrive, and had to wait outside the SouthWest terminal for a while for the shuttle to arrive. Meanwhile, my AA flight would have arrived. Anyway, it was stressful. I made the connection to an intl flight out of LAX, but I couldn't use the lounge (despite a 4 hour connection time)..

I booked a bunch of other flights with overnight "connections" (separate bookings) that went without a stress even though there were some small delays which would have allowed me to connect the same day.

Murphy's law - flights get delayed badly when you have a connecting flight within a few hours, but they are on time (or have minor delays) when you connect the next day. :/