Should an aircraft's registration match with its tail number?
The registration number painted on the aircraft is required to match the aircraft's registration certificate (and airworthiness certificate). If the numbers don't match the aircraft's paperwork isn't in order, and it can't legally be flown.

There is a procedure for changing your N number which is usually used to put a "special" (reserved) number on an aircraft:


Send the FAA a letter requesting an N number change.
(It probably helps to have a good reason - like say United Airlines buying a used jet and wanting to re-register it with a N___UA tail number from their reserved block).
Wait for the FAA to send you paperwork authorizing the N number change.
You will get 3 copies of form 8050-64 - two for the FAA, one for you.
Put the new N number on the plane.
Usually you'd do this by repainting the aircraft (or the portion of the tail where the number is), but you could also do vinyl stickers or something similar)


  One of the copies of form 8050-64 becomes your temporary registration certificate at this time, and you must carry that along with your original registration certificate showing the old registraton number.
  Within 5 days of applying the new N number return one of the copies of form 8050-64 to the FAA to obtain a new registration certificate with the new N number.
  Within 10 days of applying the new N number return the other copy of form 8050-64 to the FAA to obtain a new airworthiness certificate with the new N number.
  
  I expected to find something stating how long you have between the time the FAA grants the request to change registration numbers and when you must apply the new one to the plane, but as far as I can tell you can take your time doing this: there doesn't seem to be an expiration on the authorization to change N numbers.  




As long as you comply with the requirements for the registration number change the aircraft is only "grounded" for as long as it takes to apply the new registration number (if you're repainting this could take a while ; if you apply a large sticker with the new number on it it could be done in less than a day).