Neuropathy in the hands and/or arms in patients with rheumatoid arthritis may in rare cases cause wrist drop. "When a joint swells, it can pinch the nerves of sensation that pass next to it. If the swelling irritates the nerve, either because of the inflammation or simply because of pressure, the nerve can send sensations of pain, numbness, and/or tingling to the brain. This is called nerve entrapment. Nerve entrapment most frequently occurs at the wrist (carpal tunnel syndrome) and elbow (ulnar nerve entrapment). A rare form of nerve disease in patients with rheumatoid arthritis that causes numbness and/or tingling is neuropathy. Neuropathy is nerve damage that in people with rheumatoid arthritis can result from inflammation of blood vessels (vasculitis)." Types
Types of wrist drop are distinguished by the nerves affected:

Weakness of brachioradialis, wrist extension and finger flexion = radial nerve lesion
Weakness of finger extension and radial deviation of the wrist on extension = posterior interosseous nerve lesion
Weakness of triceps, finger extensors and flexors = c7,8 lesion
General weakness of upper limb marked in deltoid, triceps, wrist extension and finger extension = corticospinal lesion

Diagnosis
The workup for wrist drop frequently includes nerve conduction velocity studies to isolate and confirm the radial nerve as the source of the problem. Other screening tests include the inability to extend the thumb into a "hitchhikers sign". Plain films can help identify bone spurs and fractures that may have injured the nerve. Sometimes MRI imaging is required to differentiate subtle causes.
The following situations may result in wrist drop:

Stab wounds to the chest at or below the clavicle–The radial nerve is the terminal branch of the posterior cord of the brachial plexus. A stab wound may damage the posterior cord and result in neurological deficits, including an inability to abduct the shoulder beyond the first 15 degrees, an inability to extend the forearm, reduced ability to supinate the hand, reduced ability to abduct the thumb and sensory loss to the posterior surface of the arm and hand. Broken humerus–The radial nerve can be damaged if the humerus (the bone of the arm) is broken because it runs through the radial groove on the lateral border of this bone along with the deep brachial artery. Lead poisoning–Wrist drop is associated with lead poisoning due to the effect of lead on the radial nerve. Persistent injury–Persistent injury to the nerve is a common cause through either repetitive motion or by applying pressure externally along the route of the radial nerve as in the prolonged use of crutches, extended leaning on the elbows, or regular upper body rope suspension. The colloquial terms for radial nerve palsy are derived from this cause. Correcting dislocated shoulders–Radial nerve palsy can result from the now discredited practice of correcting a dislocated shoulder by putting a foot in the persons armpit and pulling on the arm in attempts to slide the humerus back into the glenoid cavity of the scapula.