Billed an extra month?

To my recollection, mobility bills are for both the past and future. You are billed in advance for basic use of the options in your base plan, and for additional charges (international long distance calls, for instance), even if they are zero. 
It also seems odd you received a bill on January 13 for the period you describe. I’d have thought it would arrive in advance of the use period, so I agree with Starchoice.

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It's hard to comment without seeing the bill and seeing what differs from the previous ones. However, it seems that you may not have had Telus service for the mentioned dates since you switched providers. It's a basic question but did you cancel services with Telus and when? If you cancelled on or around December 27th when your new service started, it seems highly unlikely that Telus would bill you for a full month starting January 12th.