Yet something in their faith - or perhaps something in the air - seems to imbue those who remain, and especially the young novices, with an irrepressible optimism; and, sometimes, their high spirits burst into the open.
In the last days of the summer, around 60 monks from the Labrang monastery in the Xiahe province crowd into a rusty bus to go on what could almost be described as a works outing. The hired 'lama bus' takes them some five or six miles along mountain roads to the local grasslands: vast, flat meadows some 9,000ft above sea-level.
They spend three days camping there, their blue-and-white tents and crimson robes making bright patterns on the grass, like wild flowers. It is an ancient festival, although it has only recently been revived after a lengthy Chinese ban, but it is far from solemn. Rather, it is a celebration of the joys of being alive. The older monks spend much of the time praying, chanting and dancing. The younger ones, who are in the majority, are less restrained. Some listen to Tibetan pop music on ghetto-blasters; others sing or play the guitar. Many are wearing football boots or trainers under their robes, and a vigorous, exuberant game of something halfway between football and basketball is one of the highlights of the holiday. The words 'Gary Lineker' are enough to bring a smile to every face, but most faces are smiling anyway. As one old man, the father of one of the monks, observes: 'Today you see my son at his happiest.'
Before the Chinese came, the Labrang monastery, which is one of the main centres of the dominant Gelukpa (Yellow Hat) sect of Tibetan Buddhism, had a population of 4,000 monks. But in the Cultural Revolution 3,000 monks were killed, and the remainder sent to a labour camp near Urumqi in Xinjiang province. Fewer than half their number returned alive. Though the monastery buildings are now being restored, the monks are still subject to persecution -two men in their twenties were arrested just a few days before the festival. Despite this, many Tibetan parents see sending their children to the local monastery as the best way of ensuring a decent education for them.
And young and old still believe that, one day, their spiritual leader will return to them from his exile.
Watching the young men hurling themselves after the ball, you wouldn't necessarily guess that they were pacifists, but they seem to be enjoying the experience. And as they play out their game against the vast backdrop of the Himalayan grasslands, with the shadows of eagles hovering over their dragon-patterned tents, you catch a glimpse of what Tibet might be if it were allowed to breathe again.