Depends how much the award is and what it'll be used for. If it is used for the same things you're funding with a Fulbright, Fulbright will (most likely) alter your funding package. For example, if Fulbright was providing 1000 Euro per month for 10 months and another ward would offer you 1000 Euro, they might deduct the 1,000 from the 10,000 total they would pay you. But it depends on the amount and what it is used for. Also, note that they can rescind the Fulbright if they found out you got funding from another source and did not report it.