How do motor capacitors change in value with respect to age or environmental factors?
Yes, capacitors such as motor start capacitors can change in capacitance value due to aging and high temperature. The change in capacitance could actually be the cause of the motor no longer being able to start but there are probably additional factors as well. One of these being ESR of the capacitor (mentioned in @KevinReid 's comment) can change (usually increase) which seriously degrades the function of the capacitor. Usually the change of ESR also relates in some way to the net change in capacitance as well. 

Other common capacitor failures include:


Breakdown of the insulation layer between the plates in the capacitor can result in a short circuit in the capacitor. Often when the short occurs a lot of current will flow and this can cause internal connections to blow open like a fuse and render the capacitor open circuit. If the capacitor design does not support this internal fusable connection type the excess current can lead to overheading and the capacitor physically exploding or catching fire. 
Over temperature operation can cause the seals on a capacitor case to fail which can be catastrophic for certain types of capacitors that have an internal damp nature due to an electrolyte. If this dries out the capacitor can lose most of its capacitance.