Bootstrap 3 has a nav-justified class that can be used on nav. No extra CSS required:
<div class="container">
  <h3 class="text-muted">Project name</h3>
  <ul class="nav nav-justified">
    <li class="active"><a href="#">Home</a></li>
    <li><a href="#">Projects</a></li>
    <li><a href="#">Services</a></li>
    <li><a href="#">Downloads</a></li>
    <li><a href="#">About</a></li>
    <li><a href="#">Contact</a></li>
  </ul>
</div>
Demo: http://bootply.com/72519
Based on the comments, to have full-width centered links using the navbar-nav class, use flexbox...
.navbar-center {
    width:100%;
    display: flex;
}
.navbar-center>li {
    flex:1 1 auto;
}
<div class="navbar navbar-default">
    <div class="container">
        <ul class="nav navbar-nav navbar-center text-center">
            <li class="active"><a href="#">Home</a></li>
            <li><a href="#">Link</a></li>
            <li><a href="#">Link</a></li>
            <li><a href="#">More</a></li>
            <li><a href="#">Options</a></li>
        </ul>
    </div>
</div>
https://www.codeply.com/go/6ZE3obnpuP
Also see
Bootstrap NavBar with left, center or right aligned items
Center Navbar in Bootstrap