Can you have more than three game files on Mario Bros. Wii U?

You should if there is functionally on the Wii U to back up save data as what there was on the Wii

On the Wii with Dragonball Z Bodukai Tenkaiichi 2 it used a global save where if you wanted to start a new game you'd end up erasing the progress of the current data and it was a big game featuring scenarios from almost every Dragonball Z Saga and Movie (to the date of the game's release) plus "What If" scenarios and scenarios where you play as the bad guys/second party.

Because it's such a large game and my brother originally brought it and played it it would be a problem for someone like me to who wanted to start from the beginning wiping out his progress and build things up my own way. In order to get around this we used 2 SD Cards, one for him and one for me. When I wanted to go on I would take his SD Card, back up his game to it then take my SD Card and restore my save game. When he wanted to play, vice versa.

Now provided the Wii U has similar if not the same kind of save game backup functionality you can do a similar thing by backing up the original data. When the players of that data want to resume their game(s) simply backup the current data on a separate medium (SD Card, Harddrive, etc.) and restore the original data and if the second group wants to play you update the first backup and restore the second.

The answer by Memor-X explains how you can have more then three savegames. I will try to explain why you cannot achieve it in the 'normal' way.

Some games have a different savegame for each account. Some games, such as NSMB-U, have a shared savegame, with different slots inside that samegame. The third option is a savegame for each Mii, such as Nintendoland. In the first case you can have as many savegames as there are profiles on the Wii-U (until the space runs out). In the second case you are limited by the amount of different slots determined by the developer of the game. You can bypass this in the way Memor-X describes.