A hard inquiry, which occurs when you apply for credit and your credit reports are pulled as part of the application process, will remain on your reports for two years. Your credit score may also take a small hit as a result of the hard inquiry, but you shouldn't see a significant downward swing in your overall scores. A drop of more than five points likely means something else has changed in your credit profile.
In addition, hard inquiries can only hurt your FICO scores for the first 12 months they are on your credit reports. After the first year has passed, they may still remain on your reports for another year; however, they will not impact your FICO scores in any way.
If you check your credit reports and find hard inquiries that don't belong, be sure to officially dispute them through the major credit reporting agencies. Request their immediate removal and provide all the documentation you can gather to support your dispute.