If a company (or person) is technically insolvent that merely means that it has a negative net asset value; its liabilities are greater than its assets. The significance of technical insolvency depends on circumstances: it may be an indicator of serious problems that may lead to actual insolvency, or it may be perfectly acceptable.
It is perfectly possible to be technically insolvent, while still being able to repay debt. It is also possible to be technically solvent and unable to repay debt. This is because technical insolvency is based only on the balance sheet and ignores cash flows. In addition book value is very different from resale value.
A simple example of technical insolvency that does not lead to actual insolvency would be an individual who has negative equity in a mortgaged property, no other assets, but an income sufficient to keep up with the mortgage repayments. A business might become technically insolvent through similar falls in asset values, or through heavy expenditure that cannot be capitalised such as research.
Conversely, a company that is technically solvent but which is no longer profitable may well become insolvent. The resale value of assets is often much lower than their book value. In some sectors, such as banking, it is often useful to consider whether a company would be technically solvent if all assets were fully marked to market.
Legally, technical insolvency in itself presents no problems (in the UK). An individual can only be declared bankrupt by a court at the request of themselves or an unpaid creditor. A technically insolvent company is free to keep trading as long as the directors reasonably believe that the company will be able to pay its debts, and, again, as long as an upaid creditor does not use the courts to force a liquidation. Of course, the state of the balance sheet does have a lot of bearing on what the directors can reasonably believe!