That does happen in some cities - also to provide an alternative to exorbitant inner city parking costs - but it will add a lot of overhead to your trip, so for a daily commute it becomes unfeasible unless you're okay with adding 30-60 minutes to it. Every day.

(I did public transit to my assignment (working as a consultant) for five years; won't work with my current one as it's a 2 hour journey for just 60 kilometers).