Yes, but not necessarily for the given reasons.
An example:
On my current project, we are building a tool for data entry. We have certain functions that are used by all (or almost all) tabs, and we are coding a single page (the project is web-based) to contain all of the data entry controls.
This page has navigation on it, and buttons to interact with all the common actions.
By defining an interface (IDataEntry) that implements methods for each of the functions, and implementing that interface on each of the controls, we can have the aspx page fire public methods on the user controls which do the actual data entry.
By defining a strict set of interaction methods (such as your 'cut' method in the example) Interfaces allow you to take an object (be it a business object, a web control, or what have you) and work with it in a defined way.
For your example, you could call cut on any ICut object, be it a knife, a saw, a blowtorch, or mono filament wire.
For testing purposes, I think interfaces are also good. If you define tests based around the expected functionality of the interface, you can define objects as described and test them. This is a very high-level test, but it still ensures functionality. HOWEVER, this should not replace unit testing of the individual object methods...it does no good to know that 'obj.Cut' resulted in a cutting if it resulted in the wrong thing being cut, or in the wrong place.