Yes, please check with your own insurance company if your insurance is valid for the whole period of your traineeship and for entire Europe. Find out about the conditions and rules for reimbursement of medical expenses as well as about the necessary procedures to be reimbursed.
Trainees based in Brussels should check if they need to register with a 'mutualité' for reimbursement.
In Luxembourg, there are insurance companies which can ensure complementarity. Note that the procedures and rules may differ from one insurance company to another as well as from one country to another.
If you have your own health insurance and you need to register with a 'mutualité' to obtain reimbursement for your expenses, you can find a list of 'mutualités' on the INAMI website for Brussels and on GUICHET for Luxembourg.