Mount Everest is a popular mountain to climb, largely because it's the tallest on Earth. It's not considered the most difficult- there are mountains in the Himalayas with higher death rates, with Annapurna rating highest- but because it's the tallest, it's the most popular. In recent years, it's been very popular. There have actually been traffic jams at the summit, as due to the weather, people have to make their attempts in similar timeframes, often having to wait for one group of climbers to get off the summit before they can take a turn.
This link, showing a photo from Outside Magazine and reprinted by the Daily Mail, shows just how bad it's been getting; the photo shows a line of 150 people, only a fraction of the 600 that were on the mountain at the time. It's as if you took everyone in line at a videogame console launch in Manhattan and plunked them on, well, Mount Everest. People seem to be getting the impression that climbing Everest is something that isn't incredibly difficult to do.
And they're starting to pay the price. Just because it's a popular mountain doesn't make it an easy mountain, and just because there are a lot of people around doesn't mean Mother Nature isn't just as ready to swat some of them off her throne. Which is precisely what happened to four climbers the day after the photo was taken, who got caught up in a windstorm and tumbled down the mountain.
The feeling among experts and locals is that people have simply lost respect for the task they're undertaking. The Guardian article linked above quotes one German climber that ultimately turned back, Ralf Dujmovits, encountering several people that never should have been on Everest in the first place, including an 80-kg (175-pound) French journalist who had burned through her oxygen supply before even having reached the hard parts, and one American climber who for some reason was hell-bent on dragging his bicycle to the summit.
The small timeframe during which one can feasibly summit, combined with the seeming ignorance and/or defiance of the death zone- the point above roughly 26,000 feet at which there isn't enough oxygen to sustain human life (Everest's summit is 3,000 feet higher than that)- means there's only so many people that can be up on the mountain at once. You bus climbers in by the barrelful, and odds are some of them aren't coming back down. Spend too long on Everest, and frostbite becomes a factor as well.
And now there's a new risk: global warming. Rising temperatures on Everest have made for unstable ice, which leads to lost grip, which leads to plummeting climbers. Switzerland's Eiger- its north face the most notoriously deadly climb in the Alps- requires low temperatures to be able to climb it, because the north face has too much loose rock to be climbable without ice freezing the rock in place. Everest may see the ice itself come loose, breaking away in big chunks and potentially slamming into an entire line of climbers there because they had to wait for the warmest weather available to make their attempt.
You're not outrunning that on a bicycle.