Not sure if your tenant is bluffing about suing you, but if you are following the law and your lease, then you have nothing to worry about. 

Not sure if I should stick to my winning bid or accept the offer.The bank might be bluffing or maybe not?

And look, they had YOU bluffed from the outset!

If you have no proof like this, it could be hard to prove in court, but I am betting they are bluffing.

Did he look into the other 2 suits and make sure he wasn't getting bluffed?

Adventurous rollercoaster. my funding fell through; seller realtor said he had just received 2 other offers (I thought he was bluffing) and listing was pulled 24 hours later - it says pending.  

(so you can be 60% sure they aren't bluffing)."

Bottom line is, sellers simply want the best bang for their buck, or he might be bluffing.

Please remember to ask for the proof of the other offer amount so you're not bluffed interested something.

I was also not bluffing and willing to walk away because I never fall in love with the property, I just know what will and won't work numbers-wise.