Re: Atlantic-Pacific-Caribbean-- there may be such a point in the Andes mountains in South America, but it depends somewhat on what you define the boundaries of the Caribbean to be. The Caribbean is really just an arm of the Atlantic. If you include the Gulf of Mexico with the Caribbean, then the watershed of the Caribbean would include the entire Missippi River basin (roughly from the Rockies to the Appalachians and from the Gulf to the Canadian border, excluding the area immediately surrounding the Great Lakes.) It would also include Texas and most of New Mexico, most or all of Alabama, and the dividing line with the Atlantic would probably run right down the Florida Peninsula (though that would be subject to interpretation.)
Eureka! I just looked at an old National Geographic map insert (Nov 1993) on “Water.” It has a little map dividing North America into drainage basins. It has the Pacific and Arctic oceans by themselves, but the Atlantic side of the continent is divided into Hudson Bay, Atlantic, and Gulf of Mexico. The western boundary of the three Atlantic parts and the Arctic section runs, of course, down the Continental Divide (call this Line 0) from Alaska down through the Rockies to Panama. I’ll try to describe where the dividing lines for the other sections go:
the 3-ocean point you heard described is in the Canadian rockies on the Alberta-B.C. border, From there a line (1) runs northeast towards Baffin Island, which is the big island that covers the entrance to Hudson Bay. North of this line and east of line 0 is the Arctic Ocean drainage basin. The next line (2)runs from about where the Continental Divide intersects the US-Canadian border and runs east towards northern Minnesota, skirts around to the north of Lake Nipigon, and runs across Ontario and Quebec to intersect the sea at the northern tip of Labrador. Everything between this line and line 1 is in the Hudson Bay basin. Another line (3) runs from near the source of the Mississippi River in northern Minnesota down to Chicago, over to Buffalo, down the Appalachians to northern Georgia, and straight down to the Florida everglades. Everything east of this line and south of line 2 is in the Atlantic basin. Everything else (west of line 3, east of line 0, south of line 2) drains to the Gulf of Mexico.
From this map, it appears there is no 4-way point. If you include Hudson Bay with the Atlantic, Atlantic-Pacific-Arctic meet at one point and Atlantic-Pacific-Gulf meet at another point. these points are about 300 miles apart. The former is in Jasper Natl. Park in Alberta (or is it Banff NP??) The latter is probably inside Glacier National Park in Montana.
See? A picture really is worth a thousand words. 
Oh, as for your Mt. Everest question: Remember that the height of a mountain doesn’t necessarily mean it is a watershed divide. Look carefully at a map of Colorado and you’ll see what I mean. Conversely, in many places you can barely tell you’re crossing a watershed-- for instance, the Continental Divide in southern New Mexico.