In your original post, you note that the bad capacitor prevents the power supply from rising past 0.3V. If you had measured this bad cap, you would notice that it no longer works like a cap, but like some sort of resistor of relatively low value. That is how you check for a bad capacitor - by checking that it is not shorted or otherwise has enormous 'leakage current'.
Checking low level leakage current is troublesome, since a Ta capacitor's leakage current will change over time after the capacitor is installed in a circuit. Once the capacitor is driven to its operating voltage, any micro-shorts that can get cleared at that voltage will get cleared and the oxide layer at the short will be re-grown. As long as the voltage stays at this level, the leakage will settle to an extremely low value. Increase the operating voltage, and more sites that could short out at this new higher voltage will then short out, until all of those are cleared. In this mode, the leakage current is the current required to clear the shorts, and once the short has cleared, no further current will flow. Because of this, once a Ta capacitor is installed in a stable circuit, leakage current will become extremely low and its reliability will increase to extremely high levels.
So, the simplest answer is that if you can power the cap up to your desired working voltage and the 'leakage' current decays toward zero, the cap is good. The leakage current of a cap that is not good at a certain voltage will increase limitlessly when biased to that voltage. Catastrophically failed Ta caps will have such a large short that they will conduct current even at very low voltages, like those from an ohm-meter.