It's possible, I do it on my laptop. I dual-boot Windows 7 and Linux, and I can hibernate one and boot into the other, no problem. The two systems shouldn't affect each other. There's a few things that might be wrong:
Are you using hibernate to disk, or suspend (to ram)? You need to use hibernate, or else you'll get the situation you describe. In hibernate, all data is saved to disk and the computer uses no power - you can unplug it and pull the battery and still resume. Suspend stops everything except the RAM, which still draws power. If any lights remain on with the power cord unplugged, you're not in S4 hibernate - you need to enable it.
Hibernate needs to be supported in your BIOS, which it should be in any computer made in the last decade or so. It also needs to be enabled in Windows. Click the start orb and click the arrow next to the "shut down" button. If something on that list says Hibernate, hibernate to disk is on in Windows and something else is the problem (check your BIOS). If something there says Sleep, turn on hibernate to disk by doing the following: Go to Power Options in the Control Panel. Click "change plan settings" next to your current power plan, expand "Sleep" and turn "Allow Hybrid Sleep" off.
If this doesn't work you might be loading the Windows bootloader before GRUB (the bootloader than Ubuntu installs), although this is unlikely because usually if that's the case you can't boot into Linux at all. If my first suggestion doesn't work, I'll write something here about this.