Yes you can, providing that there is availability in the limited upgrade “class”.

• Upgrading economy class to premium economy class

  You can only upgrade more expensive tickets, NOT the cheapest fares.
  The more expensive BA fare classes that you can upgrade from, are described
  as Y, B and H class and these are the highest economy fares.
  Example, Boston return, H class is £996. B class is £1,146 and Y class is £1,646!
  The lowest fares of say £446 at the moment cannot be upgraded.
  Example, Cancun return fares, currently only £1,281 to £1,936 fares can be upgraded

• Upgrade from premium economy to first class
  This is a better option as British Airways will allow upgrades from all the lowest fares
  (these are W, E and T class) The same applies for upgrade from Business to First;
  All fare levels: J, C, D, R, I class are included