Could Aleve cause fluttering or palpitations?

Yes, Aleve and other NSAIDs (Aspirin, Ibuprofen, and prescription-only meds like Toradol, and also the recent lawsuits against Vioxx) can cause heart palpitations, and you are taking a higher dose than is recommended (the pill insert says do not take more than 2 in 24 hours).

If pain has been a major issue for you, I'd go to a doctor and tell him/her that you have bad side-effects from Aleve, and ask for help with whatever is causing you the pain. There are much safer pain-relievers out there than NSAIDs (even Opiates are safer, when taken as prescribed).

NSAIDs have the risk of causing stomach ulcers AND heart problems. But I'd tell your doc the truth about your double dosing and follow their advice (assuming the doc is competent, which is often not safe to assume).