Is it possible to overload a panel with too many duplex breakers?

YES.

By duplex breaker, you mean 'tandem breaker', a special breaker designed to allow you to get more circuits into a panel that is full or almost full.

You most likely have a 'CTL' type panel in your home.

You MUST use 'CTL' breakers, or CTL tandem breakers in that panel. By design, if you look at the bus bars that your breakers snap onto, the upper 60 percent or so is different than the lower 40 percent.

The lower bus bars are designed differently so that you cannot install tandem breakers in a certain number of spaces. This prevents you from overloading the panel.

Most people at Lowe's and Home Depot do not know this and will give you anything to put into your panel. DON'T! If the CTL tandem doesn't fit, it's not supposed to. Do NOT go back to the store to get a different breaker because that one didn't fit. Call an electrician to install a larger service and panel with more capacity. Installing non-CTL breakers to cirvumvent the safety feature can cause your house to burn down.

It costs more to rebuild your home due to electrical fire, than it would have cost to have the job done right by a professional electrician.

The above answer is correct, but I would like to add there are some States that do not allow tandem breakers.