This is one of the toughest questions in Aikido practice in my opinion. I think it totally depends on who you are training with and the particulars of the movement each and every time you do any technique. It can be as broad as a "rank" thing, meaning I can depend on someone who has been around a while to be able to take care of themselves, to knowing you are with a beginner that you should take it very easy with. With either person we should do all practice remembering that they may not be physically or psychologically up to par. Just because Joe MyUke was able to take a high fall from this throw last night doesn't mean he wants to, or is able to take a high fall tonight. I try to start slowly and build up to faster speeds when my partner and I are in agreement, either verbally or physically.
Some folks are naturally stiff and are not flexible, try to give them a break. Some are not sure of their Ukemi, try to give them a break. Some of your partners are trying to work on being flexible. They have received feedback regarding their "stiff" Ukemi and are over compensating the compliance part of being an Uke because they are learning. Try to give them a break. Some are jerks that wish to dominate and show how good they are in comparison to you (I have not found very many of these, and if you asked me to point out an incident I would not be able to think of one), try to give them a break, and remember, you don't HAVE to train with them. Some may be intimidated by you, try to give them a break. You may often be frustrated with your training because you cannot find someone that you think is taking proper Ukemi for the throws you are trying to do. If you are training at a "young" school it will take a while for you and your partners to get the skill necessary to take the falls appropriate for "realistic" flowing motion. Be patient. I believe it is best to start slow and work your way up to fast and furious in all cases.
I have also found that the person I dread most to train with is often the one I need to work with. The reason is they are bringing up some "issues" with me personally, and I need to learn how to deal with them in an Aikido-like manner. Everyone has something to teach us (even if that "lesson" is to not train with that person again!).
The bottom line is, when we are in a class and trying to learn what the instructor is sharing, try to mimic what the teacher is doing. If your partner is not making it easy to do this, it is time to say; "Can I try it with you doing it this way this time?" Try to make your wishes and needs clear.
Aikido is an art that is very difficult to learn because there are so many variables. We are not standing shoulder to shoulder punching the air. We are not kicking the person in the head or punching them as hard as we can. When other martial arts are practiced they have to alter their techniques to make them non-injurious. Sometimes (as in Judo) they are almost purely a "sport art", one of points and fouls. We are not usually doing a Kata, because each and every time we do any technique, it is different from the one we just did 2 minutes ago or 5 years ago. This is why it takes around 5 years to get a black belt in Aikido and you can get a black belt in other martial arts sooner. Aikido is more complex. You are actually "sparring" almost every class. Some martial arts don't even allow students to practice with each other until they have been around a certain amount of time
I also believe Chuck is right when he says to "Deliver a committed, sincere attack...". Remember however, a committed, sincere attack does not have to be fast. Just clear, direct and full of intent.
With all of this in mind, we always need to pay heed to safety. One of the most powerful aspects of Aikido is the fact that we can practice full speed and complete each technique. We can practice these techniques and commit them to "muscle memory" without harm to our training partners. Most other martial arts teach skills that maim or kill the attacker, and though they are able to practice these techniques safely in class, they can never go all the way because it would maim or kill their partner. I am still working on what would happen if the person you are doing certain Aikido techniques to doesn't know how to take care of themselves (Ukemi) in a real life self defense situation.
Still learning myself.