I've heard this story before and I think it might be apocryphal. Usually that same LA Times piece is cited as well.
More likely is that 160 characters was what the GSM signaling channel comfortably allowed, essentially making it free.
Its still this way today - SMS uses an out of band signaling channel and cell phone companies charge you as data.
http://latimesblogs.latimes.com/technology/2009/05/invented-...