Based on your post, it sounds as if you have bought into some myths that have been spreading across the pages of this forum recently.
First, IPs have no bearing on search engine rankings, forget about all that nonsense. It was made up of dubious hosting companies to part naive webmasters from their money.
Secondly, SERP rankings are not based on the number of backlinks, but on the total relevancy score of the web page for the searched keyword.
Backlinks do indeed influence relevancy scores, however, they carry different weights based on PR and trust. Site-wide links do carry weight, but since they share a common set of inbound links there is only so much PR passed due to the limited diversity in the backlink graph. That limitation of backlink sources has nothing to do with IP addresses, and everything to do with a lack of diversity in the backlink graph. If a competitor has a diverse backlink graph, then more PR is passed due to that diversity, even if the diversity all comes from pages that share the same IP address.
Forget about IP addresses that is a red herring, backlink diversity has nothing to do with IP addresses.