First we should precise up what is exact same time. As mentioned previously absolutely and exactly correctly, the exact "exact" same time would be extremely rare to achieve... However we do not posses such a hardware to measure events with such "exact" time resolution and we register events with pretty huge time windows/frames. Will we get spectral pile-up or not depends a lot from the hardware: counters (EDS SiLi or SDD or gas proportional counter) response to ionisation (or Townsend's electron chain or what ever it is called in given solid or gas counters) ( that is raise and fall times of discharge/voltage drop in the counter through temporary bridged anode and cathode), used decoupling capacitor between counter (which affects how much we can saturate counter with intensive beam and not completely drain the capacitor, and how fast it resets) and charge-preamplifier. Additionally to that further in pulse-processing chain there can be shaping electronics and delayers, amplifiers and pulse-hold analog circuits to hold the max registered amplitude of pulse for synchronous (and unfortunately often multiplexed - single ADC for few signals) with used clock analog-digital conversion, so that MCA could be done (which is in digital domain and pile ups does not happen there). One of main widespread misunderstanding is that deadtime has anything to do (or causes) with pile-ups - it has nothing to do with that! They can look as related as they respond to the increase in density (in time space) of pulses, where the probability of pile-up do increase too.
Pile-ups in 99.9% of cases are caused with too slow preamplifier+pulse shaper. Old preamplifiers (from 80'es still unfortunately stuffed into some recently built EDS and WDS) outputs pulses with pulse length of ~1µs, or down to 500ns (that is enormous timespan which makes possibility to register two (and more) x-ray photons as single event much more probable), so the probability to get a pile-up (per single pulse basis) with 10kcps input rate is about 10/1000, but with 200kcps that is 2/10. Actually math of this process is much more complicated than this, as probabilities of tripple, qoudruple, quantiple and more pile-ups affects the probability. Thus I agree, that probably some simple algorithms can cope with simple-pileups, but I don't think (and had not seen) that even modern EDS systems are capable to cope with n-tiple (n>2) pileups appearing with counting rate approaching 1Mcps. As far I could find, in example Amptek's newest generation preamplifier would be able to cope with 8 million pulses a second. That is only a single order of magnitude improvement compared to technology from three decades before. That is understandable, as it is not possible to shorten the sample time much further without sacrificing the precision and accuracy of measurement. To conclude, the pulse-pile-ups are non-linear process with exponential-like growth responding to increased count rate.
Dead time - it depends (EDS counting circuit vs WDS counting circuit). WDS gas proportional counter has no dead time. Dead time in WDS is generated at analog-digital domain crossing (and queuing) explicitly. On EDS it is more complicated, as preamplifier circuit has its dead time, and analog-digital converting circuit has another. Anyway deadtime is completely linear to counting rate in both cases.
Which leads me to say that this
Search for "EDS detector dead time pulse rejection sum peaks".
is completely irrelevant to this problem as pulse-pileups and deadtime is two separate independent from each other processes, while both dependant on counting rate, but their response to that is different.
I want to emphasize to N.Ritchie, that actually that is not only not rare, but happens in significant amounts during most of everyday EDS or WDS analysis. Actually WDS is more badly affected as pile-ups are completely not dealt with at all or their existance ignored completely, and correction is tied explicitly to dead-time. Which does not work at all with very high count rates (>50kcps). But even with moderate count rates (lets say major elements at typical probe current of EPMA of 10 or 20nA), the pile-ups will be about 0.5-1.5 % of counts, which is hidden normally with over-engineered deadtime correction procedures and most of operators are not even aware that there are these pile-ups at all (unless would be such adventurous as me hanging the oscilloscope to raw signal outputted from preamplifier).