http://i.imgur.com/SOUgeze.jpg
So if you look at a map of the border between the United States and Canada, you'll see that much of it lies on the 49th parallel north. This starts in Minnesota and runs west across North Dakota, Montana, Idaho, and finally ending in Washington.
After that boundary line hits the ocean, however, it continues again and cuts into a peninsula that happens to be there. In other words, they made the boundary not continuous, thereby creating a lot of hassle for residents of Point Roberts who cannot travel to the rest of Washington by land without crossing the border twice.
What were they thinking? This seems entirely inconvenient and unnecessary. Wikipedia says that this area was "overlooked," but how do you miss something as obvious as that? This part of the border seems arbitrary and like it's there for the sake of having a border. Why not just let Canada have it?
Essentially, the Treaty of Oregon was signed before there had been any surveying done to see if setting the border at the 49th parallel would cause any problems like this. Once it was signed, they surveyed the land and realized Point Roberts would be an anomaly. The British Foreign Ministry tried to offer trading Point Roberts for some equivalent change elsewhere on the line that would cause less of a problem, but as far as I'm aware, there's no record of how the US responded--in any case, they never ceded the area.
Hayes, Derek (1999). Historical Atlas of the Pacific Northwest: Maps of Exploration and Discovery.
