From 1983 until 1990 I was working on a 5 man maintenance team for a public school system in Massachusetts that had a total of 9 school buildings. In the course of an average school year we had to change so many flourescent ballasts that we actually purchased them on one ton pallet lots. Needless to say I think I had opportunity to run the gamut of possible problems with flourescent lighting fixtures.
Ballasts can be tested with a volt meter, however you must keep in mind that a ballast is a step up transformer that converts the 110vac line voltage to a much higher voltage, typically in a range of 475vac to 730vac. While the actual amperage is quite low, I can testify first hand that the secondary wiring of a ballast can and will introduce a tremendious shock and I would not recommend live testing ballasts unless you have a real good electrical background and specific experience with high voltage transformers.
Having said that, there are a number of reasons why your fixtures may not be working besides a defective ballast.
Typically ballasts are made in two styles, single lamp ballasts and two lamp ballasts. Normally when you have a fixture with more than two lamps it will have the lamps mounted in pairs with a separate ballast for each two lamps in the fixture, I.E. A four lamp fixture will have two ballasts. The ballasts are then typically wired so one ballast will supply current to two alternate lamps. In a four lamp fixture one ballast will power the first and third lamp, and the second ballast powers the second and fourth lamp.
In order to understand the problems we must first examine the theory of flourescent lighting. You will note that there are two electrical contact pins on each end of the flourescent lamp tube. Immediately inside the tube there is a high resistance tungsten filament connected between those pins on each end of the tube. The tube is then filled with a gas that will flourese or glow in the same manner as a neon lamp when it is heated and acted upon by electron flow, thus the name flourescent tube. The inside of the tube is then coated with a white powder that emits a white light when bombarded by the free electrons flowing through the gas.
At startup the ballast passes current through the filaments, which heat up quickly and in turn heat the gas in the tube. As the gas heats up it becomes electrically conductive and the path of current through the gas has less resistance than the fillaments. The current then takes the path of least resistance passing the electrical energy through the tube, thereby dropping the high resistance filiment out of circuit automatically.
This also explains why when flourescent fixtures are operated in a cold enviroment such as an unheated garage or workshop in winter, or when the lamp is not making full contact on one end, the end of the lamps turn black because the filiment stays hot too long while attempting to heat the gas in the tube to operating temp. The presence of black marks on the glass on the end of the tube is a good sign that the filaments have been overheating and may or may not be burned out. If one filament burns out it then falls on the opposite filiment to attempt to heat the gas, and in most cases the second filiment will also burn out quickly.
Most flourescent lampholders are designed so that you install the lamp by aligning the pins in a vertical line and slide the pins into the lampholder, then turn the lamp a 1/4 turn so the lamp pins will make contact with small brass spring contacts in the lampholder.
On the bottom edge of the plastic lampholder there is a molded raised rib index line. In turn, the metal end cap of the flourescent lamp has a small dimple stamped into the metal. When a lamp is properly installed the dimple in the metal will be perfectly aligned with the index mark on the lamp holder. If not, the lamp is not rotated properly and perhaps one or more of the lamp pins are not properly contacting the internal spring contacts. When replacing lamps always carefully check the alignment index on both ends of the lamp tube.
In most instances the wires are connected to the lampholder by means of sping type push in connectors. Make sure the wires are tightly held in the lamp holder.
Before installing a new lamp visually check the lamp holder. If it is cracked or broken replace the lampholder before installing a new lamp tube.
If you install a new ballast and the light flickers or does not start quickly on the end of the fixture where the yellow wires are connected, disconnect one yellow wire from each of the two lampholders and reverse them. This will normally correct the problem.
When replacing a Ballast it is critical that you check the type and model number of the ballast to insure the replacement is exactly the same type.
). I kinda know electrical but I make lots of notes and do allot of double checking cause of the terms used. So if someone does have a test suggestion please try to use simply wording. 
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