You can use the same part. However, you will not be able to edit the part in the context of the new assembly unless you delete the in-context references to the old assembly.
Also, note that if you do make edits in the new assembly, the part will update in the old assembly (if the parts are in the same directory).
This brings up document control issues. If the same part is truly the same part (no differences), feel free to use it in several assemblies. If you make any changes, you should change the file name (Save As. ..) so things don't get confused. When you open an assembly, the parts are called into the assembly by file name.
Between major revisions, I normally will open my "master" assembly (that holds all parts and sub-assemblies) for the project, create a new directory (such as 050103 for today's date, within a project directory such as Piston Engine 01), and hit File, Find References, Copy to copy all parts referenced in my "master" assembly into the new directory. I then close the assembly and open the assembly in the new directory to make any changes for the new revision.
This practice is very basic and leaves a good document control trail. It won't work in many scenarios, however, such as multiple people accessing the same parts on a server. In that case, you should consider PLM software or some other document control practices. It's also critical to have a regular back-up process so files cannot be lost in case of a hard drive failure, fire, etc.
Jeff Mowry
Reality is no respecter of good intentions.