Yes, but usually Eurocontrol's CFMU will kick the flight plan back for the second one filed on the grounds that it is a duplicate flight number. This is usually fixed by adding on an extra alphanumeric character to the original flight number (eg. 522A, or 9522).

Standard practice is that airlines will not have the same flight number departing anytime during the same calendar day (measured in UTC / Zulu). This is what CFMU checks for when accepting a flight plan. It is only when the departure times are closely spaced but on either side of midnight Zulu that the computers may accept it and create a situation where both flights are in the same FIR simultaneously.

Also, just as an aside, the OPERATIONAL flight number (used for flight planning purposes) and the COMMERCIAL flight number (seen by the consumers) may or may not be the same. Many airlines use randomly generated operational flight numbers rather than the commercial flight number to avoid conflicts like this. So what a passenger knows as flight BA 123 may actually have an operational flight number like BAW 3X7G.