Yes! Caltrain is actually very accommodating to bicyclists, with no extra charge for bikes and special bike cars (that are often full at peak hours).

I personally bike about a mile to a train station in Palo Alto, and about a mile from Caltrain to the office in San Francisco.

However, San Francisco has some fundamental problems that make it less hospitable to biking than Copenhagen. It is notoriously hilly, and doesn't have the awesome protected bike tracks that Copenhagen has.