I'm looking at Bluehost and ANhosting. I'll be moving 6 domains, all Drupal 6.x, all low traffic. I am very concerned about the "bad neighborhood" effect as well; I've read a number of posts where moving to a new provider caused a significant drop in traffic. There are good and bad posts for both, but not much recent material.
I assume you'll be moving from a current web host? If so, and you have some time left on your account, I would recommended keeping your account with your current host (with all your website files) and just duplicate everything on the new web host. Once you re-direct the domain to the new host, it will allow you to test whether you really do see a drop in traffic. If it's too much of a drop and there's no question that it was caused by the transfer, then you can use the money back guarantee and remain at your current host.
Those were just some initial thoughts of a solution.
They are big hosts and I suppose that is normal if they have mixed reviews.
I recommend you get in touch with them tell what kind of problems you had and about your expectations. So I suppose they will be able to workout good solution for you.
I've used Hostmonster, which is basically another name for Bluehost. The beginning (almost 2 years ago) was not bad. But lately, the server has been getting slower and slower, and uptime slightly worse as well. I would not recommend them for a CMS like Drupal, which is known to not be easy on servers.
I am very concerned about the "bad neighborhood" effect as well; I've read a number of posts where moving to a new provider caused a significant drop in traffic.
I doubt this has much to do with the effect you're referring to. Search engines are very much aware that shared hosting is the norm, and that changing IPs is very easy to do. If there would be any penalty involved, it would likely have to do with things that are under the host's control.
I've been with AnHosting for almost four years. It was initially for a very small web site, but I added one a bit larger over two years ago. I was happy with the service until last year. The company moved servers over the summer and it was a disaster. My server move was botched four times, causing several outages and a big dig in Alexa/Google ranking.
I'm considering moving to a VPS, but I also have a lot of credit with AnHosting. I was thinking of using the credit to try a year of AnHosting's "Power Boost" service to see if it makes a difference. I haven't been able to find many reviews on Power Boost though. The standard hosting service is something I can't recommend.
Most large hosts are known for slow server and support speeds. So loss of SERPs are more often related to poor site speeds. The "bad neighborhood" effect has more to do with shyster sites, such as "SEO hosting" IP addresses, or other mostly spammy content being found on certain IP ranges.
Bluehost = EIG = yuck, horrible host, avoid at all costs.
AnHosting = MidPhase = UK2 = another megahost with plenty of reviews at WHT.