Question: Is there a camera that can capture all the shades of a dark foreground object as well as a bright background?

I am trying to take a picture of my black cat against a bright summer sky. My camera will only give me one of these two:

* A cat, with fine gradations of shades in the fur, against a flat nearly white background or
* A solid black object silhouetted against a blue sky with gossamer clouds.

But my naked eye will take in all the detail, whiskers and cloud.

Am I seeing an inherent limitation in any camera or just my cheap digital?