There seems to be some confusion about this, and as we recently went through this experience, thought this information might be useful to someone.
We have EU passports, and extended our stay in Tarija.
The immigration office (oficina de migracion) is on Calle La Paz close to Calle Bolivar.
Once the initial 90 days is up, it's apparently not possible to extend your stay – at least not for free. It's also not possible to nip over the border (in our case, it would have been to Argentina) and return after a couple of days. It is possible to pay for a visa which will entitle you to another 30 day stay.
To get this visa, you need:
A letter from a lawyer (abogado, there are many to be found in town) stating why you want to remain in the country. Cost BOB 100. You need a copy of this letter for each person.
Photocopies of the main page of your passport (2 copies), the original entry stamp in the passport, and the so-called tourist card (the piece of paper given to you on original entry)
A print-out of a recent bank statement showing that you have sufficient funds for the length of the extended stay. One copy per person – it can be the same bank account.
A passport-type photograph – 4x4 cm, red background.
Return to the immigration office. Once they're satisfied you've met all the requirements, you'll be issued with a small piece of paper with a bank account number, and told to pay BOB 600 per person into this account at a bank (Banco Union in our case). Return to the immigration office with 4 photocopies – per person – of proof of payment from the bank.
You should be issued with the visa the next day.
It's possible to extend your stay again after this – for another thirty days. (Not sure if this process can be repeated indefinitely, presumably not). The second time, there is no need to reproduce all of the documents, just the photocopies from the passport, including one of the previous visa, and the photograph. Cost BOB 300 per person.