Is it possible to create a custom web part using MVC?
Some developer have somehow managed to adapt the Microsoft MVC pattern to SharePoint (sample), but it would require additional work (and sometimes modification to the web.config files). Furthermore, the mvc pattern is not easily adaptable to the web part pattern SharePoint is based on.
Your best bet would be to resort to a more traditional MVP pattern, as show by the patterns & practices SharePoint Guidance project. There are also some codeproject projects that attempt to create a custom version of the mvc pattern, but -from my experience- the MVP pattern from the guidance is usually more suitable to work with.