I've already applied for one post-doc fellowship with one project with the consent of my possible future supervisor. I will know whether I've been succesfull in 6 months.
Now I realised that there is another fellowship to which I could apply - my potential host supervisor agreed that she would support me for both fellowships.
From my view I see no obstacles to ask for both, nevertheless I want to ask, what would be the opinion of a potential reviewer if he/she will get the same project for review twice from different agencies? If I will apply for both fellowships, both the review process and the results anouncement would run more or less in the same time period.