It's more common for older models to come with Java, which will allow you to run applications based on that platform.
Definitely it just depends on what version of html your game is written in and what your phones browser/interpreter will support
I have only tested on my own Samsung B2700, which is a relatively new dumb-phone, other phones may provide different results.
Testing my own phone:
While it is hard to judge the viability from my simple test I have no doubt that a simple Tetris clone or similar is at least possible, and it would probably play reasonably well.
I made the following page to test keyboard input, it is blank when loaded, but should fill basic information about keyboard events on the screen as the keys are pressed: http://ebusiness.hopto.org/phonetest.htm