The weakest part of Indian finance is probably the cooperative banks. EPFO's Employee Pension Scheme (EPS) probably has a funding gap of 1% of GDP, but the cooperative banks are most-likely worse than this. Rediff has a fascinating collection of news items about fraud and crises in cooperative banks.
From a public policy perspective, it is generally thought that customers of cooperative banks are the worst off in being helpless, uninformed depositors - unable to understand the risk of banks and unable to take care of themselves. This motives all sorts of paternalism in terms of putting down government money to rescue failed cooperative banks, even beyond the ordinary processes of deposit insurance.
A big crisis in a cooperative bank took place in 2001, with Madhavpura Mercantile Cooperative Bank (MMCB). In a fascinating recent paper, Rajkamal Iyer and Jose-Luis Peydro have examined this episode. They seem to have obtained some unique data from RBI, which isn't easily available normally.
The special feature of MMCB that they focus on is a unique mechanism for interbank contagion: many other cooperative banks had held deposits with MMCB (!). Hence, when MMCB went down, there were fears about other cooperative banks also failing. As is well known, cooperative banks have negligible equity capital, and are hence unable to absorb even the smallest adverse shocks. So if a cooperative bank had even 1% of it's assets with MMCB, this would be enough to induce insolvency.
At the time, there was a run on many a cooperative bank in Gujarat. What Iyer & Peydro find is that there was more rationality in these runs than meets the eye. They find that the contagion was significantly related to exposure-to-MMCB. Banks which had a greater exposure to MMCB experienced higher depositor flight. Banks which were more sound experienced smaller withdrawals. Depositors didn't panic irrationally.
I see this as some kind of `strong form efficiency': depositors seem to have been able to obtain information about cooperative banks which is not in the public domain, and then make judgments about which banks were sound and which weren't.
This evidence contradicts the popularly held notion in India that depositors - particularly depositors in cooperative banks - are helpless, are ignorant and require paternalistic support from the government in the form of various kinds of risk-management subsidies for their transactions with banks.
India's story with cooperative banks is far from finished - as the Rediff index page above clearly suggests. A recent, and excellent piece on this subject, by Tamal Bandopadhyay appeared in Business Standard.