It's iffy. Definitely not a "no", but the stars would have to be aligned correctly for "bullets" to become money.
Why? That gets a little complicated, but it's not beyond the ability of the average person to understand, so I shall endeavour to explain.
Money has three attributes that give it its usefulness. These attributes are:
- Measure of value
- Store of value
- Medium of exchange
So, let's look at how well bullets meet the objectives:
Store of value
This one is actually pretty good. The lead will always be usable as lead. The brass will always be usable as brass. Both materials tend to keep a fairly consistent value, all other things being equal. The powder and the primer (assuming these are modern cartridge "bullets") will last several years as long as they're not abused. If it's military-grade ammunition that's formulated specifically to be stockpiled, it may well last decades.
So, in terms of every-day commodities that could be used to store accumulated wealth, bullets aren't that bad. There may well be better choices, but choosing bullets won't result in your wealth rotting away like it would if you chose something like apples.
Measure of value
If you want to do any kind of budgeting or other monetary planning your money needs to be quantifiable in consistent units. This means that "fungibility" is an important quality. Fungibility is how interchangeable the units are. A particular ounce of 24 karat gold is functionally identical to any other ounce of 24 karat gold, so gold has good fungibility.
"Bullets" don't score quite so well as gold does, or even as well as the lead and brass they're made of do, because their chemical components decay, so newer ones are inherently more valuable than older ones. But it's not enough completely disqualify them. Mass-produced bullets for mass-produced guns are otherwise quite fungible. If they weren't, then you'd be likely to blow up your gun at some point, and that would be bad.
You do run into an issue though where not all guns shoot the same bullets, and obviously a .22 bullet isn't worth the same amount as a .50 BMG bullet. Keeping track of the exchange rate between .22, .44 magnum, and .50 BMG would be headache enough, and that's only three. There are thousands of varieties. If you live in the United States, go down to Wal-Mart at some point and peruse their ammunition section. Then keep in mind that Wal-Mart only bothers to stock about the top 2% most popular varieties, which brings us to:
Medium of exchange
This is where it kind of falls apart. Firstly, you said that bullets are very common. Well, they're also very heavy. They need to be common enough that everybody is willing to accept them in trade, even if they can't use them themselves, because they know they can resell them later, but not so common that you have to carry around 20 pounds of brass and lead just to buy dinner.
The big thing though is the previously-mentioned wide variety in modern ammunition. If your gun is a .30-30, shells for a .30-40 are useless to you, even though visually they appear almost indistinguishable. If you want it to work out, you're going to have to cut down on the variety. Otherwise tracking the exchange rates between the different kinds rapidly becomes a nightmare.
There are a couple of ways you could do this: First, you could decree that somehow, in the economic collapse say, most of the variety was eliminated and, say, only NATO standard weapons still exist. Alternatively, you could kick the tech level back down to the end of the muzzle-loading era. At that point a bullet is nothing more than a chunk of lead, and you effectively have a society that uses lead as a currency and, by custom, mints its coinage in a fashion suitable for launching out of a gun. That's unlikely to happen given how common lead is and how many things substitute for it, but it wouldn't be impossible. Also, most people in that era carried their own bullet molds so converting from one size to another would be relatively trivial, thus eliminating the variety problem entirely. So you're mostly just left with the fact that lead is really quite common, and really quite heavy, so hauling around the quantities necessary to buy things could easily become impractical. People would be highly likely to switch away from lead to the more traditional copper, silver, and gold. Those three have much better value/weight ratios.
You also had a question about how well consumable currency works. Consumability actually has little impact on whether or not something makes a good currency. To the extent that it does have an impact, it tends to make the currency more desirable because it increases the likelihood that you will be able to pass it along for something else you want, since people will actually use it directly.
If you've ever heard the phrase "worth (his/her) salt," it comes from the fact that a long time ago, in certain places, salt was used as currency. You can't live without a certain amount of salt, so everybody was willing to trade for it. In the areas in question, it was scarce enough that its value per pound was high enough that reasonable quantities could be used to complete transactions. So it met the Medium of Exchange requirement. Salt is completely non-perishable as long as you don't let it get washed away, so in areas where the production of salt is roughly equal to its consumption it was a good Store of Value. And finally salt is easily divisible into small units and pretty well completely fungible, so it made a good Measure of Value.
Of course, if you accumulated your wealth in salt, and then moved to an area with a nearby salt-mine, you'd rapidly discover that your wealth wasn't worth all that much. Which is why salt was not the only thing used for money. Things that are in universally consistent supply make much better money when dealing with large geographic areas, which is why refined metals have been a common choice throughout history. They're still not perfectly consistent, but they're better than regionally scarce commodities since they're relatively scarce almost everywhere.
So, overall, I wouldn't expect bullets to become the major currency of a society unless they are both in short supply and the variety of guns available is severely curtailed.
However: I would expect them to be a commonly-traded item since they are commonly desired, reasonably fungible, and keep their value reasonably well. In fact, I'd expect it would be rather similar to the old salt trade where communities that were far away from the ammunition factories trade for ammunition with the more generally used currency (gold, silver, sea-shells, whatever rises to the top in the wider area) and then uses the ammunition as one of their local currencies internally, similarly to how cigarettes and snack foods get used for money in prisons.