Let's put aside the LED alternative for a moment and take your initial question at face-value.
To "engineer" a solution, some math is required. Let's say you have a couple incandescent bulbs running at 16V and drawing 5 Watts total. You want to sustain a, say, 1/2 second loss of power without the lights blinking/flickering.
Compute the total energy needed. Power (watts) x Time (seconds) = Energy (Joules). Example: 5 Watts x 1/2 sec = 2.5 Joules.
The Energy (Joules) stored in a capacitor is 1/2 x Capacitance (Farads) x Voltage x Voltage. Example: 1/2 x 4700uF x 16V x 16V = 0.6 Joules.
Your experimental capacitor is under-sized by a factor or 4 or 5 times. But on top of that, as alluded to by others, a capacitor has the "annoying" characteristic of dropping its voltage as it dumps its energy. Compare this to a rechargeable battery which, relatively speaking, holds is voltage relatively constant; then the voltage drops fairly rapidly at the end of the "charge". As voltage drops, the bulb brightness dims so you need a circuit to boost the rapidly decaying capacitor voltage up to 16V DC...I'd look into an eBay "boost" voltage regulator for a few dollars (free shipping).
Peeling the layer of the onion a bit more, you cannot fully recover all the stored energy in the capacitor and there are transaction fees (losses) in converting voltages. So if you choose this route, I'd plan for a larger capacitor of, say, 10 times what you have now...or 47,000uF which could be ten 4,700uF caps stacked in parallel if that's all you have available. Obviously this could take up quite a bit of space.
So now what? OK, as RJR suggests above there are newer capacitor technologies that have tremendous energy density relative to the conventional "electrolytic" technology of your 4,700uF cap. A 1F (1,000,000uF) 5V supercap (less than $5) is about the same size if not smaller than your 4,700uF/25V electrolytic. It can only be charged up to 5V but using the formula above, that's a stored energy of 1/2 x 1F x 5V x 5V =12 Joules! That's 20 times the energy of a single 4,700uF capacitor charged up to 16V.
But now you need some circuitry to manage the charging of a supercap to keep it to 5V, and follow it with a boost regulator module to pump it up to 16V DC when needed. Again, a few eBay voltage modules for a few bucks a piece can handle the circuitry side.
So if you want to stay with incandescent bulbs, you can solve the blinking problem!