as far as I experienced, they can't
they can only distinguish japanese, vietnamese, korean, and some of their own dialect
they can only guess your nationality by your skin colour...
if you are black, you are african
if you are white, you are american
if you are yellow, you maybe a  local
if you are brown, you may be indian or mongol

Nah, I have had a few non-native speaker colleagues, most of whom spoke Chinese well enough, and no one could tell at all, even the difference between a native speaker of English, Greek, Russian, Tagalog, Sweedish, Finnish... All of them were "you america country obama you like lady gaga, etc"

Funny, just yesterday two Chinese men approached me and asked me where I was from.  After aswering "American", they told me it could not be, my skin was not "white", my hair is blondish, my eyes blue.  So, a nice conversation developed.  In America, not everyone is white, there are other skin colors and tones in between.  From African-American (black), to Indians (yellow-red ???), to whites.  And if you live in Northen States, your skin is whitish than those in Southern States, where a unit of sunligth covers more area than if living further South.  It was hard for them to understand that America is a "melting pot" of many ethnic groups, there is no sterotype American like in many other cultures.  But it did became a nice conversation with two fellows who were full of misconcptions, at least I had a good time

I'm surprised at the answers so far!

Chinese can usually tell the different between Japanese and Koreans speaking Chinese but generally can't deduce whether a Spaniard or a German is trying to master their local tongue because of their general inexperience with these people as a whole. They can hear the problems but can't normally identify whether it's due to the individual foreigner's linguistic problems or due to their native country.

In addition, most Chinese people who do get enough exposure to foreigners are usually there admist an English environment and know where people are from before hearing their Chinese.

The wife cant tell when we speak English
The Irish and Scotts are the hardest.
she cant tell Australian from Kiwi
or USA  from UK most of the time.
so not only do we all look alike we sound the same aswell

Yes they can usually tell where your from before you speak chinese to them. Even if a person speaks chinese they will still have their accent so it is easy to distinguish between which country someone is from. Just because somebody speaks chinese doesn't mean their original accent disappears altogether. In movies people who play characters from other countries may try to lose the accent for the movie and the role. In daily life people won't try to lose their accent when they are trying to get a simple message across.