Electronic – Electrolytic capacitors connected with one terminal only

This is a challenging question to answer. The best answer I can give is "caveat emptor".

You first need to know that the brand is reputable. My experience leads me to implicitly trust these brands: Nichicon, United/Nippon Chemi-con, Rubycon, Panasonic and Samxon. Major manufacturers should be able to provide life test data to justify the hours-of-life ratings that are on the parts.

You also need to know that the application isn't over-stressing the capacitor. A bad design that leads to excess ripple current or operating temperature or excessive voltage (or some combination of the three) may cause even the best capacitors on the market to fail prematurely. Using a general-purpose capacitor in a circuit that needs a high ripple, low-impedance capacitor is a prime example of where things can go bad quickly. A good cap in a circuit with good design margins should last years without issues, as the design will accept some degradation of the capacitor over time before failing.

Only when you know these two factors would you even be in a position to judge why a capacitor failure occurred.

You need to trust your supplier. The caps need to be stored in proper conditions with date-of-manufacture to determine if any forming/healing needs to be performed on them before sending your circuit out into the world - ideally, the caps should be new enough that this isn't an issue, but sometimes you need a certain value and you have to make do with whatever you can get.

Counterfeit parts are a huge problem and even a reputable shop (or international distributor) can occasionally be fooled, especially with 'lean' warehousing and part inventories moving from place to place.

I see three things that are hurting your charge pump circuit.

The capacitors are aluminum electrolytic instead of ceramic, so they have very high dissipation factor and low self-resonant frequency. These charge pumps really do require ceramic capacitors.

Second, they are radial-leaded and axial-leaded capacitors instead of surface-mount. That adds some series inductance, which tends to defeat the capacitance.

Third, the whole thing is built on a white solderless breadboard, which adds even more inductance. You'd be better off using a PCB breakout board (like a Surfboard) and solder capacitors with trimmed short leads, or better yet, surface-mount capacitors.

Not all capacitors are created equal, and not even all ceramic capacitors are created equal. The X7R and X5R types give fairly large capacitance values stable over a good temperature range. The cheaper Y5V, Y5U, and Z5V types can lose a lot of their capacitance over temperature, so sometimes we have to compensate for that loss by using higher nominal starting capacitance. (The NP0 and C0G types are highly stable over temperature, but usually are only available in very low capacitance values.)

Back in the day, ceramic capacitors were only available up to about 10uF, but nowadays there are 100uF ceramic capacitors. So the need for using tantalum and aluminum electrolytics is somewhat diminished; now it's driven by a cost vs performance trade-off.

The self-resonant frequency (SRF) is important in switching circuits, because at frequencies higher than SRF, the capacitor stops behaving like a capacitor and starts behaving like an inductor. Surface-mount ceramic capacitors are usually higher SRF and thus closer to "ideal" capacitors.

Aluminum electrolytic capacitors are great for providing a large amount of capacitance and accepting large inrush current, these are typically seen on power supply inputs. But they don't work well for the "flying capacitor" of a charge pump.

Take a look at MAX619EVKIT http://datasheets.maximintegrated.com/en/ds/MAX619EVKIT.pdf for an example of a good charge pump layout, using surface-mount ceramic capacitors.

Disclosure: I am a Maxim employee and designed the MAX619EVKIT way back when great lizards roamed the earth.