Not with 5V caps a wee tingle with a 12v one but your average modem/router will step down the voltage to 5V which most electronics use these days
capacitors are used to smooth out DC variables, Power supply voltage can sometimes supply erratic and unsmooth voltages that fluctuate greatly and if one capacitor is faulty in the chain it will effect the whole circuit.
Whilst they are used as smoothing that was more applicable to old transformer driven devices, these days it's a bit more than that and they are used for decoupling and reducing the impedance of the copper tracks due to way the chips work. So the circuit sees all it's power coming from a power source that is just a centimeter or two away from a capacitor with little distance the current needs to travel, and as each main circuit has it's own capacitors, they are effectively decoupled from each other.
You need to make sure replacement capacitors are of a low series resistance type (LSR), the lower the better, and a general purpose "smoothing" type capacitor will not suffice.