To fail during an exercise is a honorable thing. However, true exercise failure is very rarely achieved. Exercise failure occurs when you cannot complete a movement due to total neuromuscular and psychological fatigue. Picture yourself petering out on a set of push-ups and barely moving to complete the last rep. If offered a large financial incentive or threatened by physical harm, could you do 1 more rep? How about 2 more if I had a gun to your head? The mind controls the body. If the carrot or the stick is big enough you'd be surprised what you could do (not fly like Superman, but maybe lift a car off a child).
A major goal of resistance training is to engage and tire as many muscle fibers as possible. After one light warm-up set of bicep curls you will be lucky to engage 50% of the muscle within your arm. Increasing the weight, rep range, intensity and such will magnify the amount of fatigued muscle yet still not approach 100% involvement. Perhaps the most proficient way to fatigue the greatest number of muscle fibers is to work to 'failure'. This is where your level of commitment is leveraged. Are you willing or do you want to endure great muscular pain to achieve your goal? How important is this exercise goal? Do you have the proper amount of concentration and motivation to work extremely hard? This brings me to a more important question: is it even advisable to workout to 100% volitional and physical effort? I can say, without hesitation, no. Ever see those news stories where some octogenarian lifts a 1 ton tractor off her son? It may be done, but after effects on the body are astoundingly widespread; joint dislocation, burst blood vessels, torn ligaments and tendons, heart attack and more. Let's say you are not lifting tractors but working to 'failure' every time you go to the gym, the stress to your body's systems would be too great and likely decrease your training effects as you'd begin to experience symptoms of overtraining. The no pain, no gain dogma is an antiquated doctrine and has hurt many people in the gym. Training outside one's comfort zone with occasional scheduled bouts of discomfort will likely produce safe and more efficient results.
Learning to push your own limits can be very helpful when breaking through plateaus and stagnation. In order to achieve the greatest benefit from one's workout it is critical to combine the psyche and the soma. Far too many people workout with their mind somewhere else. Yes, it is better than nothing, so long they don't get injured due to inattention. If watching TV or talking on your phone while exercising helps you maintain a consistent workout regimen, so be it. Workouts like these will never produce a great athlete or lithe body. Establishing greatness takes a magnitude of physical perseverance and physiological grit. With a little more focus on one's workouts you can produce greater results in less time and optimize mindfulness skills. Research has shown that regular exercise increases cognitive functions and brain power. Einstein was fond of saying "if you can imagine it, you can create it". I'm pretty sure he wasn't talking about squeaking in an extra rep at the gym but it still applies.
Tips to help you fail
Visualize the exercise before you do it
Have a friend or trainer spot you
Use music for motivation
Write your exercise goals down and why you want them
Build your resistance to fatigue by practicing it often
Eccentric training and forced eccentrics (negatives)