Unless the injury you sustain is very serious then the chances are that within a few days you will be able to do something. By serious we're talking stress fracture, serious muscle strain and so forth.
A minor injury can be a time to strengthen up other parts of the body that are not affected by the minor injury. For example, an injury to one leg may allow the other to be exercised and of course it's likely that the upper body will be okay for exercise. In my experience there's always something that can be done as I say.
A year back I got one of my jumpers back to training after they strained their hamstring in a comp and then to compete well two weeks later. It was a grade 1 - a mild strain. This usually means a little soreness for a day or two, maybe a slight limp immediately afterwards and an inability to stretch the muscle without minor pain in the next 24-48 hours. Obviously it's best to consult with a physio to ensure that nothing else is awry. I'd read up on a protocol, I think used within football, to get players back to playing as soon as possible after a hamstring strain and I decided to use this. One day after the injury the athlete was walking and doing walking paced drills around the track as we completed about 6 laps of such easy movement. We'd do 20m of fast walking, 20 of marching... walking etc. After a couple of laps, we'd jog, perhaps just for 20m, then walk for 20m and jog again. This protocol gets blood moving through the body and this enables healing nutrients to go to the injured muscle as well as literally loosening the muscle. Muscle fibres, from what I understand, don't repair back to normal, rather scar tissue forms around the pull. I guess this is a little like superglue... it'll hold things still and can repair previously broken objects. Now that's good if the object is a vase and remains perched on a shelf... but not so good for a muscle that needs to move. So muscle fibres and the scar tissue that sort-of superglues the muscle together after an injury, can actually restrict the muscle's range. So if you don't mobilise a strained muscles, there's always potential for a further strain as the muscle can be restricted. So, by continually and gradually 'testing' the strained muscles, virtually straight away after injury you are helping it return to prior length, event though scar tissue is present.
Returning to the protocol.. each day you go that little bit faster with the drills and the jogging elements, by day 3-4 you'll probably be able to run up to 60% speed. It's at this point that progress needs to be careful as it would be all too easy to go up to 80-90% but that's when the injury is likely to return. Patience is needed, but this patience is better placed here than sitting at home and doing nothing as the injury heals itself. and potentially in a more injury prone way when it's subsequently tested
To get fully to speed takes time and I found that every other day after the first 4-5 of continuous activity then was better i.e. the athlete would move faster and faster every other day and not each subsequent day. On the day in between they would jog and mobilise still. In the specific example that I refer to the athlete was able to go flat out two weeks after the strain in the national champs and gain a medal. The comp can be a nerve wracking experience due to the strain that it will place on the athlete, but it just shows that a minor injury can be overcome quickly. Note I would not have suggested competing in a low key meet after the two weeks post injury, the third would have been a better option, but sometimes you have to if your season rests on it... Again medical advice was sought and the athlete was also receiving physio over this period.
Note: I've not mentioned icing and compression and anti inflammatory drugs/creams (Asprin is better for the first two days then ibruophen apparently..), in this post, but it's important that these be taken/administered in the optimum way to add to the healing process. A positive mind also is crucial.
Note also: these are my views and experiences, do consult a suitably qualified medical professional if you have any injury worries and doubts. Coaches have that practical knowledge which is vital for re-hab, but the medical professional should be consulted first and during the re-hab process. I've found that the exercises sometimes prescribed by physios, work for the sedentary and not athletes who need to return to action (and specific action as in movement) as soon as possible and develop a tolerance to re-injury that an everyday person would not require.