This is to confirm one of the earlier solutions (Aeric's).
My (Dell Inspiron) laptop has 3 Windows partitions (in order, 100 MB marked "DELLUTILITY", 15 GB marked "Recovery", and the main 400+ GB partition marked "OS", with all the normal operating system files and program and user data), in addition to the Ubuntu/Linux area.
Using Ubuntu's Disk Management tool, I found that the 2nd Windows partition was flagged as bootable. I tried changing the 1st partition to bootable (as suggested by Antithesis). This did not work. Following Aeric's advice, I returned to Windows 7 and (using its Disk Management tool), set the 3rd, "OS" partition to "Active". This worked! I can now successfully hibernate in both Windows 7 and Ubuntu.
FWIW, when I restarted Ubuntu I now found that the 3rd partition was marked "bootable". I infer that the Windows phrase "Active" must mean the same thing as the Linux phrase "bootable". The significant factor was not to set the first partition bootable/active, but the actual partition used for my system files.