Would a complete tear of one of the shoulder tendon cause extreme pain?

Yes: Most commonly torn is the rotator cuff, which is comprised of 3 muscles at the shoulder joint. Both acute and chronic tears can cause significant pain. In many cases, conservative treatment such as physical therapy will suffice.

Absolutely can: There are several tendons in the shoulder and 4 of the tendons blend in together to form the rotator cuff. Tears in these structures can cause significant pain under the right circumstances.