Why do callers get cut off in the middle of leaving messages?

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There are a number of reasons why callers leaving voice mail messages are interrupted or cut off while leaving messages:

Voice Mail systems are susceptible to a condition called "talkoff". Talkoff occurs when the system mistakenly identifies human speech as touch tones. Some people are more likely to experience this than others. There are adjustments that can be made to the voice mail system to reduce the occurence of talkoff. However, the more the system is adjusted to reduce talkoff, the less likely it is to detect real touch tones.
If a caller pauses for too long while talking, the system may determine that the caller has hung up, and the system will then disconnect the call.
If the volume or quality of the call is poor, the system may determine that the caller has hung up, and the system will then disconnect the call.
If there is a steady noise on the line, such as a motor running in the background, the system may decode this as dial tone and then disconnect the call.
Cell phones have a tendency to cut in and out. This can cause the voice mail system to miss touch tones, detect single tones as two tones, falsely detect hangups, etc. For these reasons, cell phone usage with voice mail is generally not supported.