That really depends on what you mean when you say "possible", and how the existing BTC address was generated.  As long as the private key for the BTC address was generated from a sufficiently random source, nobody else is ever going to generate the same number.  If the private key was generated fro something without enough entropy such as a person choosing a password, or faulty random number generating software, then it becomes much more likely that the same private key could be generated.


This is true.  However, as jordant and Amph have both attempted to explain...

2²⁵⁶ is a VERY big number.  Much bigger than most people are able to comprehend.  The odds of generating the same random 256 bit number as someone else isn't just small, it's astronomically small.  It's so small that any reasonable person would consider it impossible.


Yes.  This is wy it is very important to keep your private keys secure.  If anyone else gains access to your private keys, then they can spend or steal your bitcoins.


So, while there is only a 1 in 2²⁵⁶ chance of generating the exact same private key as was originally used to create the P2PKH address, there is actually a 1 in 2¹⁶⁰ chance of generating any 1 of the 2⁹⁶ different private keys that can all be used to spend bitcoins that are received at that address.

Fortunately, while not nearly as big of a number as 2²⁵⁶, 2¹⁶⁰ is still a VERY big number.  Big enough that there shouldn't be any concern of anyone generating any of the private keys as long as the private key is originally generated from a random enough source.