people who deposit their bitcoins in a fractional reserve institution will lose their bitcoins at some point. it is inevitable as bank runs on fractional reserve entities are not a question of if but are a question of when.
It depends on how those institutions are designed. If they are designed with contractual withdraw limits/time frames and are open about their fractional reserve, paying interest on deposits, and allowing public audits via the block chain, I see no reason why they can't exist.
Yes it can work but it needs that banking attitudes of the prudent banks of the 19th century when the gold standard was used rather than what we seen in banking today, where banks with reckless banking practices are bailed out by the central banks printing more fiat money.