Not necessarily. Generally, firms become insolvent if they can't meet their liabilities as they fall due (things get a bit more complicated for regulated firms such as insurers, which also have to worry about solvency ratios). If a firm's current liabilities exceed its current assets (where 'current' is defined as 'to be received/fall due in the next 12 months') then it's likely to be in trouble. Examples of current assets are cash, stock and trade debtors (i.e. other firms you have supplied to on credit), and current liabilities include trade creditors (firms who have supplied you on credit). These contrast with fixed assets and long-term liabilities, which aren't expected to arise for over 12 months (e.g. a 5 year bank loan).
If firms automatically became insolvent when liabilities exceeded assets, then no one would ever be able to start a firm with a bank loan, because from day one the liabilities (loan + interest) would outstrip the assets (cash).
It's also possible for a firm to have assets which exceed liabilities but still be insolvent, for example a property company with large fixed assets but little cash in the bank to meet its running costs.
If firms automatically became insolvent when liabilities exceeded assets, then no one would ever be able to start a firm with a bank loan, because from day one the liabilities (loan + interest) would outstrip the assets (cash).
It's also possible for a firm to have assets which exceed liabilities but still be insolvent, for example a property company with large fixed assets but little cash in the bank to meet its running costs.