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I'd wager that one reason is by the accumulation of mutation. From a perspective of time - a cell or organ that is older simply has had more time for damaging mutagens and UV radiation to take their toll on the DNA. It's more probable for an older cell or organ to have such mutations occur in a vital part of the genome, which may render defenseless against certain pathogens. One would think that such mutations would be selected against, and weeded out given a decent amount of time. This is not always the case. In the context of an organism - evolutionary pressures are weaker as the organism gets older, because it also becomes less likely to reproduce. Detrimental mutations build-up simply because they are no longer passed on, and thus cannot be selected against. These mutations, which manifest themselves when the organism is old, persist in the gene pool. This approach incidentally comes from Medawar's proposed theory on why we age.
kma230 gave a very good answer. I'd like to speculate that an aged organ has less ability to repair itself due to lower numbers of stem cells.