Hi Roger,
If the status light is not flashing when you download, that basically means it is not seeing any file transfer.
On the problem PC, can you try just deleting all the files from the mbed and hitting reset (nothing should run), then copy a binary from your desktop to the mbed and check the status light flashes as it transfers. A binary to try can be found at:
Take a look in explorer and check the files are appearing; unplug and replug to ensure it is not a caching issue; some older versions of windows cache USB disk files until you say "eject" or equivalent, so check if you need to do this.
Perhaps also try copying on a different binary and check it replaces it; main thing is to cut out anything to do with the browser.
Other things which are easy to forget when changing machines would be something like the downloaded file going to the wrong place (e.g. the download folder, rather than the mbed). The date aspect has already been mentioned which is a good one to check, but should only impact which binary is loaded, not whether you can store the binaries on the mbed.
If you are still having problems, you could try formatting the mbed drive and then power cycling it (unplug, replug), or even try a different USB port if it is something to do with the power available, but I suspect it is something to do with the copying process.
Simon
I noticed my PC was not downloading programs to the LPC1768 when I compiled. Naturally it ran the last program loaded. Tried all the usual switch off etc. no change. I moved to my other PC and found I could load and run a program ok. I moved back to the original PC and found the LPC1768 was running a different program. I repeated the process to double check. So, on one computer it runs one program, on another PC it runs a different program without resetting etc. Any ideas?