Hello Kees Internet,
I was able to reproduce this as well.
What I suggest you do to resolve this is to add a SafeCracker File fieldtype. In the Control Panel go to Add-Ons->Field Types and then install “SafeCracker File”.
Now go to your channel fields Admin->Channel Administration->Channel Fields and then click on the desired Channel Field Group.
You will want to create a new Channel Field and make sure the Type is SafeCracker File.
I say create new NOT edit or change because you want all the other existing image to still show on the front end. From now on when folks go to upload, they will use the new SafeCracker File Upload.
The code will change a bit.
{exp:safecracker channel="dossier" return="/dossier/overzicht" entry_id="" error_handling="inline" include_jquery="yes" id="dossierform"}
<h3>uploadtest</h3>
<p> {custom_fields}<br />
{if safecracker_file}<br />
{display_field}<br />
{/if}<br />
{/custom_fields}<br />
{/exp:safecracker}
The documentation on the SafeCracker module is here just in case you want to read up on it. I am sure you have seen it already.
Does this makes sense? Let me know if you need any clarification or have more questions.
Cheers,