I love plums, especially plum tarts, but have also read you should be careful with them.
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Side Effects

While there is no doubt that plums are a nutrient-rich fruit with a lot of health and medicinal benefits, there are some side effects of eating large amounts of plums as well. Plums are among the foods that contain significant amounts of naturally occurring oxalates, which may cause kidney stones. When you consume too many plums, the oxalate levels may become concentrated. These interfere with the absorption of calcium in the body and as a result calcium begins to precipitate in the kidneys. Ultimately, this may cause formation of small or even large stones in the kidneys and the bladder.

This risk of developing these kidney stones is higher among those who do not have a very healthy digestive system and those who do not chew their foods well. Ordinarily, physicians would not discourage you from consuming plums, but if you have a predisposition of developing kidney stones, or if this is a hereditary problem in your family, it is best to avoid consuming plums in large quantities. You may still be able to consume them and enjoy them in smaller quantities during the summer months when they are available.
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This was taken from here: http://www.diethealthclub.com/health-food/plum-health-benefits.html