Silje
 Can you tell me what version of our software you are using and on what operating system (MacOS, Linux, Windows)? I assume you are using the latest release (v1.0.20171204)
   https://github.com/rordenlab/dcm2niix/releases
I would suggest you look at the wiki to see how to disambiguate output filenames
   https://www.nitrc.org/plugins/mwiki/index.php/dcm2nii:MainPage#General_Usage
For example, consider the conversion
   dcm2niix -f %p_%t_%s ~/mydicoms
in this case, the output files will have the output filenames including protocol name, study time and series number (e.g. "t1_20170620161913_11", "fmri_20170620161913_12, "fmri_20170620161913_13") - a situation where you are unlikely to have files overwriting each other. On the other hand if you ran 
 dcm2niix -f %p ~/mydicoms
then only protocol name will be used in the output filename, making overwriting likely (e.g. in the example above both series 12 and 13 had the protocol name "fmri" so you will get "fmri" and "fmria").