In the US and many other nations, programmatic overlap of awards is generally to be avoided. This is clearly articulated by the NIH in their Grants Policy Statement description of Other Support associated with their Just-in-Time procedures. It is perfectly fine to submit funding applications to multiple sponsors for the same body of work, however, it is important to recognize that if your wildest dreams come true and multiple funders want to offer awards – you can only accept one of them. While it is tempting to submit the same body of work to multiple funders to get around the problem that the work to be done will cost more that is practical to ask for from a single sponsor, that needs to be done by figuring out how to divide the work into smaller definable components and then seeking support for discrete portions of the work from each sponsor.
Craig Grimes, who was a professor of electrical engineering at Pennsylvania State University until 2010, pleaded guilty to fraud charges stemming from accepting grants from the Department of Energy and the National Science Foundation to fund the same research. Additional details of this case were published in Nature on Februay 7, 2012. apparently, the charges again him also include misappropriation of NIH funds. Other similar cases related to scientific overlap have also generated penalties. Predictably, Congress is getting in on the action; the House held a hearing on Jan 24th in which representatives questioned whether agencies were sufficiently evaluating applications for potential overlap prior to making awards.
This is the kind of thing that could easily work its way into agency OIG work plans in future years and result in additional requirements beyond the current expectation that investigators disclose any overlaps and that institutions do not accept multiple awards for the same body of work.