I certainly do...
...there are (I believe) 2 varieites of elbow problems frequent in climbers:
1) Straightforward elbow tendonitis - I don't actually think this is particularly common. It is characterised by a definite point of tenderness on the bony prominence of the elbow; This problem, although can be more likely with poor shoulder alignment/mechanics, really has nothing to do with the shoulder. It should get better with the usual R.I.C.E. management.
2) Elbow pain that is characterised by generalised lateral arm pain, difficult to locate any point of tenderness. It generally comes on with squeezy/mantle-y moves (Font' elbow). This pain is caused by lack of stability at the shoulder causing impingement of structures that cross the shoulder joint, referring pain further down the arm. And this is the key - although the pain is located close to the elbow, there isn't any injurous processes occurring at the point of pain. All structures (Nerves, muscles,bones) can refer pain along their length - people with knee replacements will often have quite severe ankle pain, for example.
With nerves, this is even more pertinent. Take a lumbar disc bulge - it will likely refer pain down the sciatic nerve, which may only be felt at the calf or ankle. It is the same with the shoulder. The radial nerve runs along the outside of the arm - and if this is aggravted as it crosses the shoulder joint, it will cause pain further down the arm.
Thus, treat the cause of the problem and in this case, the origin of the pain.
Incidently, this problem is often lazily 'diagnosed' as brachialis tendonitis.