This is one of the many things that need to be monitored by a professional. I have often seen deals fall apart with FSBO's. In this market, even we as professionals have to dig deep to advise sellers in making the right decisions. I rarely see loan approvals 2 weeks before closing these days unless the closing date is 45 to 60 days out or more. If all you wind up with is sleepless nights and much anxiety, you can feel blessed. Hopefully this FSBO decision does not cost you time, $1000's of dollars, mitigation or a law suit dragging out for months to come. Always use a professional. The old legal saying applies here as well - 'He that represents himself in court has a fool for an attorney'. Always use a professional. It is well worth it for many reasons. All the best.
How can we know? It could be legit, it could be a stall, it could be that they are scrambling to get approved and don't really stand a chance. Delays happen all the time, but without someone knowledgeable on your side of the transaction to ask the right questions, and read between some lines, it's just not possible for anyone here to know.
Why are they not approved by the original date? What issues are causing the delay? Can they satisfy the lenders concerns or are they starting over with another lender, or loan type or are they just throwing a hail mary pass hoping something comes together?
Part of an agent's job is to vet the pre-approval letter, the lender, the buyer's qualification and capacity. Taking your home off the market for an extended period can cost you a real buyer. If the market changes while your tied up with someone who won't qualify, you may lose value due to less interest or higher rates or changes in the local economy.
Even the best agents run into similar troubles, but they have experience to help them sort it out quickly and advise the seller's what would be in their best interest.
Let us know how it turns out.