If you are wondering if the host that you use can make a difference in your SEO results, I believe I have the answer. It definitely does.
I recently had one of my SEO clients move their sites to my server from a popular "budget" host. I took over SEO on these sites about three months ago, and had been struggling to get rid of the many bad links his previous SEO provider had set up for his sites. But, because they were being punished by Google for those bad links none of those sites were ranked in the top 100.
To make it easier for me to work on them, and as a free service to my client, as a thank you for his SEO work, I moved him to my server. Since we are also planning to add a full merchant services API for our payment portal, I also gave each of them their own IP address.
Within a week, the sites showed a massive improvement in the SEO rankings, including several page one rankings for highly competitive keywords. My client was thrilled, as was I.
All I did was move the sites to my server, do some basic "on page" tweaks, and I saw almost immediate results.
I believe the "clean" IP addresses must have helped. It is possible that hundreds of other sites, sharing an IP address at that budget host, had degraded the rankings of all "associated" sites. Google claim this doesn't matter, but I don't know how else to explain the massive improvement in ranking that all of my client's sites received when they were moved to a "clean" high end server.
And, of course, I am happy to see these great results. This client had two of his sites blacklisted by Google, so to see them with first page search positions is thrilling. I'll take great outcomes from whatever method works.