Can I have the same project as another master's student (same experiments and results) and we both write two different theses?
Bring this up to your university and be open about it. Make sure your supervisor knows you are seeking extra guidance. This is very sketchy but not necessarily bad.  If you think about it, some grad work may rely entirely on secondary sources or data collected by others.  The challenging part here is that you have the same project, not just the same data.  Working together on the experiments isn't the problem - I had some masters students work with me on my experiments - writing the same thesis &quot;in your own words&quot; is the problem.
If you are both going to draw the exact same conclusions and do the same analysis, then there is an issue.  You both need to be able to distinguish between your specific contribution and your specific methods.  For instance, just having a data set in common doesn't mean much if you analyze it for property A using method X and your partner looks into property B using method Y.
To be more concrete, you and I might run the same survey, and to save resources we run it with the same participants, but you are looking at socioeconomic status as an indicator of academic success and I am looking at a correlation between socioeconomic status and academic preparedness.  We will look at different things, probably have different literature reviews (at least partially), and definitely make different conclusions.  No worries here.  We'd just acknowledge that the survey was run cooperatively.
The problem is that, during your defense, you have to be able to defend what you personally did.  You can give credit where it's due, but if it turns out that your thesis is only half yours, then giving the degree is going to be difficult.