I did that once - book connecting flights on two different tickets - and although I ended up actually making the connection (despite the first flight running 2-3 hours late), the experience was so nerve-racking I swore I’d never do that again. It helped that the flight I was connecting to ended up running late too.

The reason I did it was it was a personal side-trip (to see family) on an international work trip, and although my employer would let me mix work and personal travel on their booking (provided I reimbursed them for the cost of the personal portion), the bureaucracy involved had put me off going down that path.