You're all partly right. For a given pressure or voltage the flow is determined by the total resistance. The resistance comes from adding up the resistances of all the parts of the line in series. Narrow necks can be the dominant contributor, but they aren't the only contributor. Especially for laminar flow of fluids, where the resistance goes as the inverse of the fourth power of the pipe radius, even a short narrow segment can make a big difference. For electrical current, the resistance only goes as the inverse of the square of the wire radius.
Definitely there's a big problem with even a small narrow segment in an electrical circuit, due to the heating, proportional to I2
R, where I is the current running around the circuit. If there a small segment with high R, lots of heat gets dumped in that small region, which creates obvious serious safety issues.
I guess some of the beauty of having a mathematical way to describe these things is that it lets you assign importance to the various parts without switching into all-or-nothing verbal approximations.
(published on 05/03/2011)