Cash flow problems are probably leading to higher costs because the company has to borrow money. Also, the cash flow problems might be a result of higher costs, because if you are spending more money on materials you might have less cash. It could be either one causing the other. I suggest a careful cash-flow analysis and finding a way to borrow money cheaper, such as with a business credit card or line of credit.

If the cost for making each piece of product remain constant, then profits will remain constant no matter how many pieces are sold.

The general idea is that costs of construction for each piece should go down with increased production. This cost decrease would come from a possible renegotiation with parts vendors so that a company could promise to buy more of Part A per order and thereby get a better price. This savings then represents an increase in profit for the company. 

It could be that the firm is experiencing rising costs.  Perhaps the increase in sales has put a strain on the firm's manufacturing capacity.  Because of that, the firm might be operating inefficiently in order to fill all the orders.  Costs rise, so profits don't rise.