We have the genes for it, but they remain dormant after shutting down shortly before birth.  Within four years, we could see a mouse regrow a "finger."

   
...there is increasing optimism that therapies can be developed in the next five to ten years to prevent the formation of scars and to restore damaged or lost tissue from wounds, heart attacks, spinal cord injuries or Alzheimer's disease...

...it would probably take a year or more to grow a new arm or leg.

   Currently, there are more species with regenerative capabilities on Earth than not, humans being of the exception.  Of course, if they do regenerate human tissue, science will then embark on speeding it up.