If your assets are worth less than your liabilities, you're technically insolvent. If you can still pay your bills from cashflows, you don't need to claim bankruptcy, but on a long enough timeline without a significant change, you will go bankrupt. There was a long stretch of time post-GFC where this was the case with GE.
Banks have to 'mark-to-market' their debt periodically, but since GE wasn't a bank, they didn't abide by the same regulations. Say there were two identical houses on a street with $500k mortgages, but the market crashed, the houses are worth less, and only 1/2 of the mortgages will be repaid. If Bank of America owned the first house, the mortgage asset (it's an asset to the bank) would be marked-to-market and what was formerly a $500k asset would now be worth $250k, and the bank would show a loss of $250k.
GE doesn't have to do this and could carry assets at their previous value (and didn't have to mark down bad debt either). To simplify the story, during the GFC, GE had that one mortgage "worth" $500k and debts worth $400k. If they were a bank, they would have had to admit that the mortgage was really worth $250k, and they would've gone into receivership.
If not for that accounting quirk and the massive amount of bailout cash they took from the Federal Reserve / FDIC, they would have actually had to declare bankruptcy. Ironically enough, the FDIC program and Federal Reserve are to shore up banks and GE was explicitly not a bank per the above few paragraphs. So make sense of how they got access to bailout money solely authorized to bail out banks..
[1] - $45B in losses buried on GE's balance sheet: http://www.businessinsider.com/henry-blodget-living-on-plane...
[2] - GE borrowed $16B from Federal Reserve: http://www.propublica.org/article/general-electric-tapped-fe...
[3] - FDIC to back $139B in GE Capital Debt: http://dealbook.nytimes.com/2008/11/12/fdic-to-back-139-bill...
[4] - GE Borrows $59.3B from FDIC program: http://www.bloomberg.com/news/2010-12-01/ge-borrowed-16-bill...