Human cells typically consist of 23 pairs of chromosomes. Telomeres are repetitive sequences of DNA located at the ends of chromosomes. During cell replication, a number of basepairs are lost from the end of the chromosome and this shortening restricts the number of divisions that a cell can complete before it becomes senescent, or non-replicative. In this paper, we use Monte Carlo simulations to form a stochastic model of telomere shortening to investigate how telomere shortening affects normal aging. Using this model, we study various hypotheses for the way in which shortening occurs by comparing their impact on aging at the chromosome and cell levels. We consider different types of length-dependent loss and replication probabilities to describe these processes. After analyzing a simple model for a population of independent chromosomes, we simulate a population of cells in which each cell has 46 chromosomes and the shortest telomere governs the replicative potential of the cell. We generalize these simulations to Werner's syndrome, a condition in which large sections of DNA are removed during cell division and, amongst other conditions, results in rapid aging. Since the mechanisms governing the loss of additional basepairs are not known, we use our model to simulate a variety of possible forms for the rate at which additional telomeres are lost per replication and several expressions for how the probability of cell division depends on telomere length. As well as the evolution of the mean telomere length, we consider the standard deviation and the shape of the distribution. We compare our results with a variety of data from the literature, covering both experimental data and previous models. We find good agreement for the evolution of telomere length when plotted against population doubling.