As everyone has stated, 2 forms of protection are necessary when shooting. PERIOD. I have shot next to many a person with a break on their rifle and when that blast starts going sideways, you need as much protection as you can get.
I would also like to extrapolate on something, someone else said hinting at actual levels: when you see the NRR (stands for Noise Reduction Rating) that is the absolute maximum reduction when using with a perfect fit. Not all ear plugs fit perfectly. The expanding foam earplugs are some of the best plug-type hearing protection you can get without going custom (which are amazing and worth the money if you will be wearing for long periods of time, imho) because they conform to YOUR ear canal. Which are almost more individual than fingerprints. Usually the max a dense, moldable, foam earplug will reach is some 35+NRR. Next you need to put over-ear-type protection. Again, you must have a "perfect" seal around your ear to reach the NRR stated on the box/bag/specs. So hair, glasses, weird facial contours all make a difference. The maximum NRR I've seen on muff-type hearing protection is in the low 30s(but that's just my research ymmv).
The thing is, you can't just add the two together. They work in conjunction, but the absolute maximum NRR is a function. Google it up, there are quite a few resources and Wikipedia actually does a fair job of explaining it as well. The reason you should use both, not just one or the other, is the way hearing works:
Sounds travels in waves due to molecules in the air knocking into one another. However, those molecules also knock into everything. Some of your hearing comes from the bones/tissue surrounding your ear/the ear itself. The plugs do a really good job at knocking down high frequency stuff, but the muffs help to knock down some of the lower frequencies that actually come through the side of your head.
Anyways, hopefully that is an overwhelming amount of information and should prove to ANYONE who shoots, indoor/outdoor, break/comp/homemadepieceofmetal, that you NEED to wear 2 sets of the best ear protection you can afford. Regardless of how long you've been shooting. If you can spend $500+ on a rifle/shotgun/pistol, you can spend $100 on good muffs and foam moldable ear plugs. If you shoot a lot (more than the average joe), I would highly recommend looking into custom molded ear plugs.
(EDITED: for completeness)