Every time I talk to people from the Tai Chi circle, most of them will say, "Let's push hands."
In fact, this is a polite way to challenge someone or to test your own skills in Tai Chi.
Sure, it's true that most of the great Tai Chi masters are all excellent at pushing hands. It's also true that all of them can fight, but that's a long time ago....
Of course, the number of Tai Chi practitioners fighting in tournaments has sharply increased in the last ten years. However, they all fight like they have been trained in Muy Thai or Jujitsu.
I once asked William C.C. Chen, one of the best students of Professor Cheng Man Ching, whether you needed to be good at pushing hands to be good at fighting.
His answer was simply "NO."
He also told me that he was once beaten in a tournament when he was still a teenager, even though his pushing-hands skills were among the best in Taiwan.
"You still need to learn how to fight." said William Chen.
In a nutshell, while it's true that learning to push hands is a major part in understanding the art, you also need to immerse yourself in a variety of fighting scenarios to develop your true ability in Tai Chi. This has helped me so many times since I was a kid.