Sure, you can, but the supercap needs to be recharged.
Larger caps will present higher leakage. A cap of several Farads or under is still manageable. But even at 5.5V a 1F cap only stores E= 1/2 C V^2 = ~15J
Compare that to a CR2032 3V coin cell. Assume 2.5V average cell voltage with 200mAh. Then that's 0.5Wh, or 1800J.
So if your electronics will spend that coin cell in 3 yrs (1.6J per day), you need to charge that [on average] each day. If you use a lower storage voltage to avoid step up/downs to the supercap, you may need a slightly larger cap. E.g. a 1F at 2.5V will only store 3J. But that's still in the ballpark for the 1.6J/day, plus some buffer to account for variations in daily charges. Although you'll need to run some math and possibly a small model to see if it can really keep up.
If you intend to keep charging the supercap 'infinitely', you may need to implement a voltage monitor circuit though, and a way of sinking some current to drain the cell to prevent overcharging. Try to dimension it such that you can always out(dis)charge the supply.
Regarding your other post with the RF harvester; consider that such power levels look a lot like what is available in RFID environments. UHF RFID can operate in the 900MHz band, and typical excitation powers allowed is up to 20dBm. The incident power at several meters is then still about -10 to -20dBm. This is also a typical limit at which efficient RF energy harvesting can occur. Below that power level, it becomes increasingly harder to bridge a diode voltage drop, even with high-Q resonant filters to swing up the voltage in trade-off current (high source impedance). In addition, at -20dBm the available power is only 10uW, and even at 50% efficiency, would only yield 5uW output power.
The very low range of Vcap does seem like a design limitation. It would require quite a large cap, which has a (proportionally) higher leakage current.
In an application where I'm using backscatter sensor nodes without batteries, the communication side of my network relies on a similar device as the Powercast to create a CW excitation field. The RF frontends will need to operate at power levels of around -30 to -10dBm (optimal around -20dBm) to operate well. However, I've also looked at RF energy harvesting, but deemed it in practical for now. At -10dBm incident power from a 20-30dBm source, the range is really poor (couple meters at most), and with -30dBm incident power, the available power at RF is very low. State-of-the-art papers I looked at only showed harvesting efficiencies of 10-20% IIRC, so that 1uW available power becomes negligible.
But YMMW if your application is happy to all be clustered within the couple of meters from the power source. Practically, as I highlighted with the CR2032 coin cell, battery-free is mostly useful if replacing batteries is prohibitively expensive (labour hours) or difficult (installation details).