Depends on the carrier.

With United, you have virtually the same chance as a top level elite. Only 1K members get access to special inventory, and that's only coach inventory. You have the same access to premium cabin awards as every other member. Your lack of status won't do you much good if you have to waitlist for an award -- and it's unlikely to get special treatment shaking loose award inventory when none is available -- but even without status you'll do quite well.

Continental availability... it's poor even if you're a platinum. I feel sorry for someone who isn't.

I understand Northwest makes additional inventory available for Platinums...

I've found that American has excellent award inventory, even for me without any status. Honestly they've always offered me the most flight options of any carrier. I only represent a single data point, but I've never been shut out checking award availability.