Can an electronic system be built with absolutely no single point of failure?
Theoretically, it would be possible if somehow we could detect every possible fault (which we cannot - see next paragraph). 

The primary problem is complexity; every time we add a component, we add complexity to a circuit. In the case of an autopilot (for example) there may be several hundred (or more) components for a single channel and it becomes time prohibitive to analyse every possible fault. If we tried to detect every possible fault, we would exponentially increase the complexity of an already complex system (thus increasing the potential for faults) and then there is the conundrum of how do I ensure the detection circuits are not faulty? 

(I actually encountered this problem where the startup behaviour of an opamp caused the fault monitor to trip even though no fault was present; thankfully this was while we were still in the development phase).

We very quickly end up in an exponential loop with no apparent exit.

The other problem is that some faults are not directly detectable - think things like bypass capacitors failing open.

There may also be a (previously) unknown fault mode for some components (I had to analyse why apparently perfectly good new dry tantalum devices were failing in a spectacularly pyrotechnic manner - it turns out that the act of reflow can damage the part and if it is fed from a low impedance source, it can go into thermal runaway).

In that case, the device becomes a short circuit for a short period of time (a second or so) and then becomes an open circuit and pretty much undetectable in most cases.

Even flight safety critical equipment, such as flight control computers (fly by wire) which are redundant (in the case of some aircraft, triple redundant with the associated sensors also triple redundant) may have backups.

The communication between processors is also galvanically isolated to minimise the possibility of an electrical fault in one processor from affecting the other processing channels (but it is still non-zero).

For civil programmes that use microprocessors, the architecture of each processor is different and this is to guard against a particular type of common mode problem; a bug in the microcode. Different architectures have a vanishingly small chance of having such a bug at the same point in a programme (but it is still non-zero).

On some aircraft, the euphemistically named Alternate flight control computer is in fact a backup for the primary computer(s) and it is the 'last man standing' (I salute any test pilot who performs the test flights to prove it actually does operate as they must deliberately fail the primary flight controls).

Safety analysis is a numbers game; for level A avionics (system failure is catastrophic) the requirement is to show that the possibility of failure is &lt;\$10^{-9}\$ per flight hour as it is simply not possible to consider every single potential fault.

We do look at all circuits using a fault tree and extensive analysis (and strict derating in many cases) but the fact is that although we can mitigate against single point of failure mechanisms, we cannot state that such a mechanism does not exist; we can only state the likelihood of such a mechanism existing and fail in a predictable manner where possible.