Assuming standard atmospheric pressure as seen on Earth, and with distance above sea level as your measure...
Altitude sickness is going to be an issue. From Wikipedia's link:
Although minor symptoms such as breathlessness may occur at altitudes of 1,500 metres (5,000 ft), AMS commonly occurs above 2,400 meters (8,000 ft). It presents as a collection of nonspecific symptoms, acquired at high altitude or in low air pressure, resembling a case of "flu, carbon monoxide poisoning, or a hangover". It is hard to determine who will be affected by altitude sickness, as there are no specific factors that correlate with a susceptibility to altitude sickness. However, most people can ascend to 2,400 meters (8,000 ft) without difficulty.
The Andes mountains have supported terraced farming. This is described as a vertical archipelago.
The quechua zone refers to relatively warm, relatively low valleys falling between 2,300 and 3,200 m (7,500 and 10,500 ft). This area shares its name with the Quechua people and languages and was especially sought after for growing maize.
The suni zone rises from 3,200 to 4,000 m (10,500 to 13,100 ft) and is suitable for the production of native tubers and grains such as quinoa, kaniwa, and kiwicha. Given the innumerable valleys and micro-climates of the Andes, over the millennia Andean farmers developed over 1,000 varieties of potatoes, as well as other tuber species, such as mashua, ulluco, oca, and achira.
The puna zone (above the treeline at 3200 to 3400 meters and below the permanent snow line at 4500 to 5000 meters) is composed of high, cold grasslands, suitable largely for pasture by camelids, the domesticated llama and alpaca, as well as the wild vicuña and guanaco. The former were used as not only as pack animals, but also for their meat and wool. Vicuñas and guanacos, though undomesticated, were used for their fine and much-prized wool. Little agriculture is performed in the puna, though in the Bolivian altiplano intensive agriculture was possible through the use of waru waru raised bed agriculture, which used specialized irrigation techniques to prevent frost from destroying crops.
The montaña zone is humid and forested. Populations here were not as large as in other ecozones, as the plants grown in montaña areas were generally speaking not food crops, but rather tobacco and coca. Just as the puna is used to collect resources from wild animals as well as domestic ones, brightly colored feathers were collected from wild birds in the montaña, such as macaws.
So based on that, it looks like agriculture gets difficult above the 4500 meter mark, since that's permanent snow hight. But that also depends on latitude. The snow line is around 4500 meters near the equator. But in the Himalayas, it is closer to 5700 meters. The Alps, it is around 3000 meters.
The interplay of altitude and latitude affects the precise placement of the snow line at a particular location. At or near the equator, it is typically situated at approximately 4,500 meters (or about 15,000 feet) above sea level. As one moves towards the Tropic of Cancer and Tropic of Capricorn, the parameter at first increases: in the Himalayas the permanent snow line can be as high as 5,700 metres (18,700 feet), whilst on the Tropic of Capricorn no permanent snow exists at all in the Andes because of the extreme aridity. Beyond the Tropics the snow line becomes progressively lower as the latitude increases, to just below 3,000 metres in the Alps and falling all the way to sea level itself at the ice caps near the poles.
I'd keep your islands averaging below 2400 meters for the snow line and altitude sickness (with possible hills/mountains rising higher). Rainfall shouldn't be an issue based on elevation.