Having more than one single <h1> tag is bound to prove counterproductive. Pages were intended to focus on one main topic and the <h1> heading was created to fulfill that purpose.
While it's certainly permissible to use more than one, for example, to break a page into sections, each one with its' own <h1> heading, it would stand to reason that the more you use, the less relevant they become.
It's commonly accepted as good coding practice to use the <h1> for your main page heading, use <h2> for section headings within a page and use <h3> for subsection headings.
This provides a much more accurate representation of hierarchy within the page content and, on a larger scale, across the site as it's indexed. Big picture, creating your pages in this fashion allows you to specify which words and keyphrases are most important to the SE's as they scan the page.
Code:
<h1>Here's my main heading</h1>
I write some content that I feel might make for a great opening paragraph.
<h2>Section 1</h2>
This is the first section
<h3>This is the first subsection</h3>
<h3>This is the second subsection</h3>
<h2>Section 2</h2>
This is the second section
<h3>This is the first subsection</h3>
<h3>This is the second subsection</h3>
For the most part, you can achieve the same thing using an ordered or unordered list but it wouldn't likely carry the same worth as the heading tags.
.02