Like in may things, your mileage will vary. You just have to look at the individual game. Some older games use some or all 16 bit code. While Windows 7 64 bit can run 32 bit code, it will not run 16 bit. Windows 7 32 bit is still backwards compatible with 16 bit code.
As long as it is not a 16 bit code issue, sometimes running the the game in compatibility mode works. Sometime it doesn't. Another solution, if you happen to have a copy of a version of Window 7 32 bit or XP laying around is downloading VMware Player and load up the OS and game in a virtual machine. Most old games will play just fine like this (or if you have Windows 7 Ultimate or Professional, you could try XP mode, but I have always had better luck with VMware).
What will really drive you crazy is when a game will work on one 64 bit machine but not another, with no rhyme or reason. Probably has something to do with difference in display drivers, etc. interacting with older versions of DirectDraw.
VMware is how I have dealt with the issue, but maybe someone smarter than me will have a better solution.
Just an addition to the above... If you have Windows 7 Professional or Ultimate they come with a license to run "XP Mode" which is an already configured version of XP running in a Microsoft VM (as opposed to VMWare). So in this case, having the more expensive version of Windows means you already have access to the tools you need at no additional cost.