Explain how it is that a business which is making profits can have difficulties in meeting its debts?

Include examples.

Include examples of businesses in this situation.

There are several ways in which this can and does happen.
The most common way is that businesses have something called accounts receivable. These are the people and businesses to which they have sold goods and services but for which they have not yet received payment. On the other side of the ledger they also have something called accounts payables and payrolls. If the cash inflo is not sufficient to meet the cash outflow, even though they are profitable, then if they can not borrow the money to meet their payroll and accounts payables they may be thrown into default. Nearly every business relies on short term borrowings to meet their cash requirements when their cash inflow is too slow in coming into the coffers. "The check is in the mail, problem".

The 2nd way in which they might have difficulty is that perhaps in the past they had borrowed a lot of money which is shortly due to be paid. The company can show a profit on the books, but may not have enough money to pay the upcoming debt that is due. And if credit is difficult to obtain, as it currently is, they may be forced into bankrupcy because they can not pay their debt.

real simple

too much cash for non-deductible expenses being taken out of the business

I think this happens in a rapidly growing business. The profits need to be put right back into raw supplies.