If the capacitor is open-circuit, there will be no capacitance at all.

If it is just weak, there will still be some capacitance, but not enough to do its job properly --- in your particular case, not enough current will pass to start the motor.

The kind of capacitors used for motor starting (usually high voltage and reasonably high capacitance) tend to suffer from loss of capacitance with age, as they may 'dry out'.

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Note added at 19 hrs 49 mins (2005-09-12 08:03:33 GMT)
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Normally, if the capacitor is open(-circuit) or weak, the compressor will NOT start...

Of the suggestions in your question, 'insufficient' is probably the closest synonym, and 'discharged' is INCORRECT in this context

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Note added at 1 day 5 hrs 24 mins (2005-09-12 17:39:12 GMT)
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Note in reply to Nik On-Off's peer comment response:

With all due respect, I think my answer differs from your own in 2 crucial respects

1) You failed to address the key issue that this is an AC rather than a DC application

2) The key factor that may help Asker's translation is that 'weak' here effectively = 'low in value'