The answer is Yes. I had been told by bankers and realtors that to get an NIE, you had to have an address, and to buy or rent a vivienda you had to have an NIE and a bank account. And you have to have NIE and address to get a bank account.
All turns out to be false. But because I had been told that, I went to the Comisaría to ask. I knew no one in the area to ask to "loan" me their address. The lady I spoke to told me to use the address of the hostel I was in.
While waiting for the NIE, I signed a lease for a piso without mentioning a bank account or NIE. And then I opened a bank account without the NIE. I switched the water and electricity to my name without it.
To get it, I did not have to have a visa or photos or a home contract.
Two more "impossibilities": Most Spanish consulates say that to get a visa you must show them a contract to buy or lease a vivienda. I could not find that requirement in Spanish law, and a lawyer told me it is not a requirement. (And they will "explain it" to the consulate if I hire them). It also is constantly stated that you have to apply for the visa at the consulate near your "official residence." The law is written as if assuming you will do so, but it doesn't explicitly require it. And one person told me that they did their application in Madrid and got the visa.
UPDATE: make the answer “yes” with qualifications. In Spain, the law doesn’t mean much unless you actually go to court. The law says they cannot deny you a NIE unless you are in “irregular situation” which apparently means overstaying your legal visit. They denied me by assuming I had overstayed. BurgosAcoge wrote a three-page letter quoting various laws and pointing out the easy-to-read evidence of my legal exits and re-entrances and demanding they correct it within five days (the law also requires the process not take longer than five days).
Two or three weeks later, I went in to inquire and got “I don’t know why no one has looked at this.” Two or three more weeks, I was informed by e-mail at closing time that it was approved. They were nice enough to stay open while I ran to a bank to pay the fee, but then they said I had to come back in a few days to get the actual number. The next day was my last legal day, but I did get the number on my next trip.
Short version: the answer is yes according to law, but unless you’re willing to make an enemy demanding your rights, it could be “no” in practice.