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To be more accurate, in the vast majority of cases ESR greatly increases, and capacitance 'may' fall slightly - usually the capacitance is still within tolerance yet the capacitor is totally useless. If you're trying to find faulty electrolytic capacitors then it's ESR you need to test.
tantalum "dry" electrolytics often fail short circuit, but tantalums aren't used much except in military/aerospace equipment. i see the occasional leaky or shorted aluminum electrolytic, but the most common failure is high ESR. my experience has been 85% high ESR, 10% leaky, 3% open circuit, and 2% shorted. they're not exact figures, and that 2% should have "other failures" included. Sprague Atom capacitors with the red epoxy seal usually fail intermittent open circuit. if there's a black ring in the epoxy around the + lead, replace the cap, because the black ring is from an internal high resistance connection, and if it hasn't failed yet, it will soon.