You could, if the system is designed for it. Lots of little digital "coins" that are shuffled around between different accounts, with traceable history.

Dollar bills have serial numbers, so it is already possible to track incoming and outgoing transactions to an extent.

The problem would come in assigning a static serial number to a pool of bits. If I spent an electronic dollar, did that dollar come from the few bucks my neighbor gave me for gas after he borrowed my lawn mower, or from my paycheck? It's not like I care which one gets spent, since they are the same, but how would you determine which one get priority. I guess the same argument applies to dollars sitting in my wallet, but its something one would have to consider designing such a system.

>The problem would come in assigning a static serial number to a pool of bits.

I don't really see it. There are cryptographic protocols designed just for this kind of things. The real thing is it would be more awkward than just cash.

If for every dollar you spend, you have to be aware of its history ("Oh, sorry, I can't buy cigarettes now, I only have two research-grant-dollars and three from my non-smoking mom, neither of which may be used for tobacco"), then the money ceases to be money (in the meaning of uniform tokens of value) and you're back at some later stage of bartering economy.