Simple question
Is it possible to connect multiple accelerometers (preferably 4) to a Arduino Uno R3 board?
If so how is it done?
I want to connect accelerators to different parts of a machine that I have built for university in order to study how the different parts act independently of each other
Any help would be appreciated
Simple answer 
unless you specify which sensors there is no meaningful answer possible ...
Yes, but ...
You might need extra hardware, you might need to scale up to the faster Arduino Due.
http://arduino.cc/en/Main/arduinoBoardDue
Most libraries for digital sensors are developed for just one sensor.
When using analog sensors, you need 12 analog input. The Leonardo can do that, but you still need to get that data to the computer.
http://arduino.cc/en/Main/arduinoBoardLeonardo
What kind of acceleration sensors did you have in mind ?
What kind of sample rate is needed ?
For example to measure mechanical vibrations of a combustion engine, the Arduino can't handle 4 sensors at high sample rates. You would need a special computer plug-in card and use Labview or so.