Of course, and the opposite is also true. Players can live without a COTM. I suppose both you and me can live without playing at all. But this is not the point, isn't it?
Now, back on track.
Players still interested in Civ3 are not kiddos (well, there's at least one pestering the general discussion forum, but this is another story). They're die-hard fans, for the most part grown-up people, with a family, a job, responsabilities and such. They love to play but can't dedicate too much of their time to the game, for obvious reasons.
With this in mind, a monthly schedule is likely to be too tight, because two games per month would be too much for the average Civ3 player. People would have to choose what game to play. Worse still, people will try but fail to complete a game in time, and possibly lose interest in a competition they feel they have no time to be part of.
The best solution, IMO, is a GotM and a CotM every two months, with monthly release dates. That is, every month a game is out, being once a GotM and once a CotM. For the duration, i think 6 weeks is the ideal. Games would still overlap but not too much, and a buffer zone of two weeks would give enough time to update the scoreboard, put out the relevant threads and prepare the next game without being forced to rush things.