I had also been getting lots of "Not Responding" and VM memory low alerts, plus the computer was VERY slow - taking minutes to respond at times and absolutely awful from Hibernate or Start, I grew older waiting.
My memory was only 447 MB of RAM.
What was happening, was that my memory usage had crept up to a level exceeding my RAM capacity and Standby could not function. When selecting Standby, the computer could not comply through lack of RAM and simply shut it down.
Very briefly, Standby stores all the data in RAM whilst Hibernate saves the data in a specially created file - hiberfil.sys and is stored on the hard disk, it does not use RAM. Hence my stored data was > RAM and Standby failed, the computer shut down. Hibernate still worked as all the data was stored on the hard disk and not in RAM.
I have recently had my memory increased to 960 MB of RAM and 2.71 GHz - approx. double the previous RAM capacity.
It is worth other users noting this when their Standby is giving similar trouble. Before considering the more complex reasons why Standby has failed, check that your RAM is sufficient to deal with your memory usage. If not, you must increase your RAM. Extra memory increase is not at all expensive, quite cheap actually.
This simple and rather silly solution avoids unnecessary trouble and expense in dissecting your computer into a pile of debris and reinventing the wheel. The web is full of users with Standby problems, especially with Windows XP, but not exclusively and many of these problems could well be a simple matter of needing more RAM.
All is now working superb and my Start-up, Standby and Hibernate times are now very rapid. Never been better.
Much faster on web selections also. So far no "Not Responding" signs or memory alerts. Delighted.
Hoping this tip is of use to somebody.