Any form of vibration which interferes with the operation of a sophisticated astronomical telescope is undesirable. Ideally, the problems which could give rise to vibrations should be anticipated and corrected prior to, or during, the construction stages. However, in spite of all precautions, vibration problems have been experienced on certain telescopes after completion. Usually, at this stage, it is difficult to solve such problems and often the solutions are complex and expensive. This presentation assesses the levels of vibration found in telescope installations at the La Palma Observatory in the Spanish Canary Islands, and attempts to qualify acceptable magnitudes for these and other installations. At one of these telescope installations on the ROQUE DE LOS MUCHACHOS, certain conditions of operation arose in which vibration proved to be a problem of great concern. Initially, a number of in-house attempts were made to find the cause of the problem but it was soon realised that professional consultants with advanced equipment were needed to analyse the situation. This resulted in a team of V.C.I. consultants being invited to assist with the project further. The problem was re-examined by the consultants of V.C.I. and a new approach, which is related in this appraisal, was employed. This paper, therefore, reviews a series of special trouble-shooting vibration tests which are designed to specifically highlight problem areas so that corrective solutions and remedies can be accurately deduced. Although certain of these vibration tests have been established for other purposes, it is believed that many of the techniques discussed in this paper are novel in relation to their application to observatories.