Fumes coming in with top up...??? Is that normal

Hi, Everyone -- I just bought a beautiful 2001 Miata. I test drove it on the side streets with top down, and top up. No problems. But when I drove it home on the freeway, top up and vent CLOSED, I got a horrible headache from fumes. Granted, there were lots of trucks so hopefully it was just a fluke. Since I've never had a convertible before, I'm not sure if it's a common problem for fumes to come in even with the top up. I hope not! Only had it a couple of days, would hate to have to sell it (I am prone to migraines induced by chemical smells). Another option might be buying a hardtop and having it installed... but does that really seal as tight as a sedan?

Any insight would be greatly appreciated. If fumes just seep in when the top is up, I might as well sell before investing any more (car needs brakes and oil flush).

Thanks!
StaciLayne

Have you checked your exhaust system for leaks/holes/rusted out?

Yes, before I purchased it the mechanic checked it out. So, you're saying that's not normal? I just wasn't sure how airtight a softtop usually is. Thanks!