Is it possible for a company to show positive cash flows but be in grave trouble?

Yes in two situations it happened first one is selling inventory but delaying payable in second situation revenue goes in the pipeline not forward.

 

Yes, temporary positive cash flow.
Easiest answer the company is selling its long term assets and the source of the cash flow in this case will be investment activities, however, cash flow from operations (company core business) could be negative.
  Hope this helps!   Mohammad