Can I Increase Rent on my New Rental Property?

We often talk to Denver real estate investors who have recently purchased occupied rental property. It is not uncommon for these new investors to want to immediately increase the rent charged, in an effort to align the rental rate with current market rents. Is this allowed? If you purchase an occupied investment property, are you allowed to immediately increase the rent?
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[@musictosoothe]

If a tenant who's been renting at lower rents decides to leave at the end of their lease, can you then increase the rent to market rent when you're advertising for a new tenant? (Does Colorado have any limit regarding rent raises between tenants?)