Question: Can I license different parts of my code (different classes) under different licenses?

E.g. I have downloaded and modified a class which licensed under the BSD 3 point. 

Can I then say, file AObject.class is BSD, and BObject.class is LGPL (or MIT, or MPL, or PD, etc etc)

Answer: 
You can license code you write with whatever combination of licenses you please. However this may not make sense.

For example if you license one file with the GPL and another with a closed source license no one is going to be able to use your code as a whole. People will be able to take your GPL code and use it separately in GPL solutions but since the GPL conflicts with closed source licenses they could not use both.

you can also release code under two or more licenses. So you may release code file.cpp under both GPL and BSD and allow people to choose which license applies to them. Or you could charge people a fee to have the code with one license and provide the code under a GPL license for free.

Remember that your license choice will affect your ability to take code "back into" your project. So if file1 is Closed Source and file 2 is GPL in the same project, you will not be able to put other peoples GPL code into your GPL file as this would be a violation of the GPL license.
The issue of license compatibility aside, you can certainly license different portions of your code differently and this is quite common for commercial software that has Open Source integrated into it. One project I was on included the [mingw](http://www.mingw.org/) compiler as part of the installation as well as [ACE](http://www1.cse.wustl.edu/~schmidt/ACE.html), [OpenSSL](http://www.openssl.org/), and several other bits from around the net. It took our lawyers weeks to be certain we'd dotted our i's and crossed our t's with our legal obligations, but in the end we shipped and even made the principals of those other projects aware of our use of their software.