It is not a problem to handle even 1000s of links daily with proper automated solutions. But from my experience - under 100 links per week is very safe. More is
bad in the eyes of Google. They will surely doubt that this is natural.
MSN and Yahoo are rumored not to be so tricky about the number of links, but SEs can change their attitude in future.
BUT
Are you sure that all that 20,000 links of your competitor use proper anchor? Maybe you need to fight a much lower number of links, if you do proper anchoring.
Simplified example: competitor can have 100 links but with different anchors, and out of that 100 only 5 links with 'super bla-bla' in the anchor text. This means that you need to have 5+ links, but exactly with 'super bla-bla' in anchor title.
PLUS
It is not only about the links (and surely proper marketing of your site), but also about onpage optimization. Your site title should have 'super bla-bla', meta with 'super bla-bla' (meta is not super important, still...), the copy on the pages should better have 'super bla-bla' and better in the first paragraph of the text. Also 'super bla-bla' would be ok in domain name and/or in the URLs of the site, like
www.your-website.com/super-bla-bla-articles
King of the hill is an interesting game, but you have to play it creatively, sometimes with out of the box ideas.
P.S. And of course the remarks of other members are useful too. Make them rush to link to your site. Than you can even forget about possible problems with Google for too fast growth. If Google mods come to your site and see real value on it, they will not venture to penalize it.