What Would Happen if Everybody Flushed Their Toilets at the Same Time?

Well, the real problem is in the water, not sewer pipes.

In a new building in Memphis, strobe lights and klaxons were installed in all the office spaces and rest rooms. During the first fire drill all the alarms went off over the entire campus.

The strobe lights caused the automatic toilets to all flush at the same time - and continued until the alarms were reset.

The problem was the hydraulic pressure created by water moving through all of the pipes at top speed (limited only by the pressure from the source and drag from the pipes). When all the flushing stopped at exactly the same moment, the water pipes burst.

@The other Don: Third, pipes would not burst. ....

Unfortunately, pipes DO burst. Thankfully, this is rare, but it can and does happen from time to time.

That said, many cities have older sewer systems that have bricked in channels, not concrete pipes. These will obviously be a weak point in the unlikely case that everyone flush their toilets at the same time.

the real problem would be treatment plant capacity. It would probably accept and discharge all of the inflows, but it would not have adequate time to treat the waste.

In case of too much intake, the system wouldn't even try to feed it to the treatment plants - the dirty water will just be shunted to sea untreated.

Sorry folks, but most of the reasoning here is bogus. Let's start with the article. If everyone flushed 3.5 gallons at once and it all overflowed at once, each house, apartment and store would have 3.5 gallons of water on the floor. This looks like a lot, but in a 5 by 8 bathroom, it would be less than 0.2 inches (let alone if it spread out across the house.
Second, if everyone flushed at once, not all of the fluid would zoom through the pipes at once. If you lived five miles from the treatment plant, it would take your flush over an hour to reach the point where mine, next door to the plant, entered the same pipe location.
Third, pipes would not burst. They are designed for worst case pressure, encased in compacted earth, and attached to manholes. Each manhole has a lid that would pop off before a pipe would burst.
As for velocity, yes the fluids would gain some speed, but not whole lot. Friction losses would probably offset any increase in head pressure.
Next, the lift stations are designed for the maximum theoretical flow of the inlet pipe. It is possible that backpressure from the outlet could affect the flow rate, but, again, the worst that might happen would be an ugly puddle right around the area of the pumping station.
Lastly, the real problem would be treatment plant capacity. It would probably accept and discharge all of the inflows, but it would not have adequate time to treat the waste. The result might be a decrease in water quality at the outlet of the plant.
Nearly this scenario exists each year at the beginning of halftime for the Super Bowl. We haven't had floods yet!

@Justin-
Oh man... The stuff you learned about velocities just increasing higher and high is completely whack. If that were the case, a full bathtub would drain as quickly as a half full tub. There is some increase in the drain rate from the added head, but ultimately the size of the limiting orifice (which could be the pipe itself) establishes the restriction.

By any chance, did you learn your physics in a public high school?

Fair enough. But the point still remains. Even if it was only water, the pipes are most likely not designed to handle that much water pressure, and would therefore burst.