Depends on the cat.  My cat doesn't even try.  Others go crazy at
certain things, like the sight of another cat, and try to get out.

I fixed the problem by replacing the plastic screen with metal screen.  Some
may try others may not even bother.

Celeste

We've had some kittens climb the screen and we found the hole they made
in it before they got out.

The only adult cat we've had try to escape is a stray we took in off the
streets. He roamed the house and meowed incessantly -- and he broke out
threw a screen one night. He was an extreme case. The other adults have
all been fine w/screen.

If your cat seems a little too interested in breaking out, you could put
hardward cloth over the mesh.

Rhonda

We had one, many years ago, that chewed through a screen window when
she was in season.

But that was only one determined cat. I recently adopted a cat that
was previously outdoors. He pawed at all the windows and doors for the
first week or so. He looked so sad. But he gave up, and even though we
leave the front door open with just the screen, he has made no effort
to attack it. He just stands up with his front feet on the screen and
looks out.

It's possible, but not typical. Just keep an eye on your cats, and
take action if you need to.