Can I cut a 240V wire half way?
As long as the connections are properly contained within a box, and the box is accessible, there should be no problem doing what you want. However, cables are usually run with very little slack in them, and you'll need enough extra wire to make the connections within the box.  If you can't get enough slack in the line, you'll have to install two junction boxes, or replace a bit of the cable. 

Make sure you... 


Use a large enough box.
Use the proper sized connectors.
Use the proper sized wire (if needed).
Make the connections tight enough.


Things to watch out for...


Aluminum wire.
Even though a larger size wire has to be used, the price of aluminum wire is often less than copper.
Electrocution.
Make sure you turn the breaker off before you begin, and test to make sure the power is actually off.