Motivated by the study of annual temperature extremes, two new results on the limiting distribution of block maxima of random variables with varying upper bounds are obtained. One gives a generalized extreme value distribution as the limit, but with a different shape parameter from that obtained when the bound on the random variables does not vary. The other gives a limiting distribution that is only a generalized extreme value in certain cases. Both results consider triangular arrays of random variables in order to mimic the property of an upper bound that changes slowly with the day of the year, as seems to occur for temperature data at many locations. An analysis of 140 years of daily temperatures in New York City shows mixed results in terms of the ability of the theory presented here to provide new insights into the behaviour of extreme temperatures at this location.