A really time-efficient way to workout is to perform exercises to the point where you are unable to complete another repetition with good form. In the fitness industry this is commonly known as reaching momentary-muscular-failure.
In the gym setting this can be safely achieved using well-designed machines but can you achieve the same effect with bodyweight exercises?
Absolutely!
But some work better than others.
Push ups and pull ups are two examples where you can safely and effectively train your body to the point of momentary-muscular-failure. Eventually, despite your best intentions, you simply will not be able complete another repetition.
The key ingredients for taking a bodyweight exercise to momentary-muscular-failure are:
Being able to fail safely
Low skill and coordination requirements
Controlled rep speed
Continuous tension on the working muscles
Push ups, pull ups (and various other bodyweight exercises) tick all these boxes. Low skill requirements, easily controlled rep speed, continuous tension on the working muscles and once you stop moving you can simply lower yourself to the ground without shifting dangerous forces onto other joints and muscles.
Gymnastic exercises, explosive exercises and jumping exercises to failure… not so safe!
Weak biomechanical positions, extreme ranges of motion, high skill requirements, large spikes in force and an uncontrolled breakdown in technique make these bodyweight exercises a bad idea to max out on. If you do choose to practice them… finish with plenty of reps to spare!
So yes, you absolutely can train bodyweight exercises to momentary-muscular-failure. I highly recommend it as a super-simple and time-efficient way to train.
Just be careful which moves you choose!