No. Page Rank, which comes from backlinks, is just one of a great number of factors that determine your position in SERPs. So the correlation between links and position will not be as obvious as that. Backlinks can help, but they are a small part of a much bigger picture.
There are a lot of different factors that Google take into account that determine where a website appears in the results. One of them is keyword selection, for example if you target a keyword phrase no-one searches for (like “Lemon coloured zebras skate boarding in Paris”) then you can probably get first for it without a single link, however the more competitive the keyword, the more links you’ll need.
For more competitive keywords the quality and relevance of the links is a lot more important than the number of links, however typically you could often get to position 20-100 depending on the keyword with only 10 or so links, but would need many more than this to get to 1st position (the higher up in Google’s ranking you go, the more work is required to gain that next position).