The big problem with reincarnation is that it cannot be fully vouched. You cannot measure the soul (of course if it exists) and say "it's one and the same".
Take the most famous "reincarnated" persons in the world, the Lamaist tulku.
From a strict theological point of view (pedantic moment number one) all but a few Buddhist schools of thought deny the existence of a soul in the strict "Westerner" meaning. There's some kind of vital energy (the exact concept is very hard to explain in simple words) which at the moment of death will go onto contributing towards a new life. Mystics, sorcerers and holy men can, through their rigorous psychic training and supernatural "powers", give a shape to this energy, allowing it to retain at least part of its consciousness and allowing it some kind of choice in the matter of body to be reborn into.
This is either done to complete some kind of quest (for example reaching the samsara by methods different from the so called Short Path) or out of compassion for his fellow men. This later concept is alien to some of the most primitive and rigorous Buddhist sects, like the Theravada, and has been deeply influenced by Hinduism, like the concept of the avatar. For example the Dalai Lama is believed to be the reincarnation of an avatar of Chenrezigs, the deity of compassion.
Simple folks and the low-ranking clergy do not bother with this high philosophy. For them reincarnation is just what it is for us Westeners: the soul of a departed coming back to life in a different body.
Since many tulku in Tibet held positions of great wealth and power (there isn't just the Dalai Lama: many high ranking ministers and heads of important and wealthy monasteries were supposed to be reincarnations) you can imagine that many families struggled to have their children recognized as reincarnations of the late holy men (and women). That's where clerical bureaucracy stepped in, devising a number of trials for young lama wannabes and a number of other rites.
In my personal opinion most of these trials are absolutely no proof of reincarnation, no matter what enthusiastic supporters of Lamaism say. The appointment of tulku is either rigged for political and economic reasons or clever clergymen have devised a way to recognize bright and intelligent children to be trained to positions of importance.
It's curious to note that the belief in reincarnation is not limited to eastern religions. The Romans belived in it, probably under the influx of Oriental mysticism (see for example Virgil's Aenaid) and in some Talmudic traditions the Patriarchs are believed to be a long string of reincarnations starting with either Adam or Abraham.
To be honest with you Ian I am still waiting for some interesting instance of reincarnation: until now they were either poorly investigated, downright frauds (for example using informations freely available in history books) or cases of multiple personalities.