Yes - it will run on just four cylinders - something that happens frequently when the under valve cover harness gets loose on one side. You need a few things to start. Properly timed fuel delivery, sufficient RPM to fire the injectors, and heat (generated from compression / glow plug system)
Does your wait to start light come on? Are you getting any smoke from the tailpipe when it cranks?
This motor uses the high pressure oil pump (HPOP) to generate the force to activate the injectors. The HPOP uses a signal from the Injection control pressure (ICP) sensor on the front of the driver's side head to determine pressure, which is controlled by the Injection pressure regulator (IPR) - any of these going bad will cause a no start. Read here for more:
The cam position sensor (CPS) frequently goes bad in these trucks. Most people carry a spare. They are about 25 bucks and would be a cheap first try at getting it to start.
Check your glow plug system as well. The tests are free. Read here:
Hard/No Start? Check here first. - Diesel Forum - TheDieselStop.com
I think the best investment you can make at this point would be to purchase a good scanner - AutoEnginuity with the Ford Package - which will monitor all the parameters needed to start and run. It could save you hundreds of dollars as opposed to just guessing and throwing parts at it.
While you are waiting for that to come, I would plug the truck in for a few hours(the cord is on the driver's side of where your front license plate is) and try to start it jumped with good cables to a strong batteried vehicle. You have to have at least 150 cranking RPM to fire the truck. You won't get that with marginal batteries.