If it's the same console, then yes.
If it's online or system link, then no.
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No
Yes, It Is Possible To Play Old Xbox Games On An Xbox 360, However, You Cannot Play Xbox 360 Games On An Original Xbox.
With a modified xbox, this is possible. Either hardware or software mods will make this possible. Otherwise, NO! Also, doing so is illegal.
Not on the same xbox/ps3 no unfortunately.
No. But, if you soft mod your Wii, you can play old platform games through an emulator.