Many people will claim that heavy partial reps using a load above one's 1 rep max is beneficial for building strength, but I don't think this is the case.
The claim is that since one uses more weight he will gain more strength but the reason he can handle more weight is due to mechanical advantage, it is not that his muscles produce more force, in fact they produce the same amount of force as normal reps of the same duration, so is there really any benefit to them?
I suppose it depends on what you mean by heavy partial reps.
I've found that training the initial deadlift pull (think, from the floor to about knee height) helped my deadlift. This is because being just shy of 2m tall, I've always found the initial pull from the floor a lot tougher than the lock out. If I can get the bar past my knees, then I've got the rep.
You could count that as a partial rep (and I always trained them heavy, though not heavier than my deadlift 1RM).
I've also found that squat holds at greater than 1RM (where you load the bar up with more than you can squat, unrack it but don't squat, maybe just kind of wiggle a bit) have helped me squat more weight.
I've read from Dan John that this can have a strengthening effect around the core musculature because you're having to support the extra weight. I'm not sure if that's the case, but what it did do for me was help with the mental block I used to get when the weight on my shoulders went above a certain point.
Again, I'm not sure if that counts as a heavy partial, I used to move slightly with the weight on my shoulders, just to get used to the feeling, but didn't do anything approaching a full squat.
Lastly, look at the Olympic lifts. If you look at a full clean and jerk, then I'm pretty sure every successful weightlifter has trained the clean portion of the move with weights heavier than they can jerk overhead. The clean is often seen as a lift by itself, but you could argue the same for a lot of "partial" movements if looked at on their own merits.
Partial range of motion exercises like quarter squats are really only good for increasing strength within the partial range of motion.[1] (With isometric exercises being a particularly extreme case of partial RoM, known for causing joint angle-specific strength increases.[2]) They are inferioir for building overall strength and inferior for causing muscle growth.[1] They may however have some benefit where there is a demand for strength only in that reduced RoM, for example quarter squats may result in greater increases in jumping and sprinting ability than full squats, but studies are conflicting.[3][4]
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Honestly, partial reps are best saved for the end of a set of full reps when you want to squeeze in a few more gains. This really depends on the workout though, I've particularly found them more beneficial with body-weight exercises like pull-ups and push-ups to increase you set count or try to expand your reps in the first and average set you can handle.
For weights, this can apply as well, but you have to be more careful with weight you can't fully lift in the first place as the injury potential is obviously higher from the very beginning. In the long-term, I wouldn't recommend partial reps as it trains poor form and reduces visual tone gains of weight lifting. Strength and bulking will benefit but if you want muscle definition and tone to accompany this workout, do the full reps with solid form and slow steady pace and breathing. You can gain muscle definition along with strength and mass if you do fully flexed reps with proper form.
Overall, lifting weight that is beyond your capabilities of 10-20 full reps in 3+ sets will just hurt long-term progress, missing out of full movement, form, etc. You'll gain in some areas, but you'll adapt to partial reps and eventually the ignored part of the full reps will grow weaker. Technique within the lift is arguably even more important than the weight too - proper form, slow solid movement, complete mental focus and flow, sharp awareness of any excess pain, stiffness, injuries, or anything else out of the ordinary. I personally didn't start gaining more strength until I did the lifts slow and fully, studying the form and putting my ego aside and having the humble humility to known where I stand and inquire feedback from the pros.
It depends entirely on what you're doing and what you're trying to achieve by doing it.
For example, a block pull deadlift can help people get passed a sticking point in regular deadlift. It helps train your CNS to holding super-heavy weight. Coupled with the fact that you are physically holding the weight, so it should most certainly help training the regular deadlift when you lower the weight back down.
Then there is something called the squat walkout in which you load a lot of weight on the bar (110% or so), unrack, then just stand there for a few seconds while braced. Then rerack. It can
Then of course with Olympic lifts, it's common to train parts of the lifts. Like a snatch-grip deadlift is technically a partial of a snatch. It can be performed at a higher weight than an actual snatch. Which has carryover to the full lift.
