Github: Can I see the number of downloads for a repo?

In Github, is there a way I can see the number of downloads for a repo?

Solution

Update 2019:

API /repos/:owner/:repo/traffic/clones, to get the total number of clones and breakdown per day or week, but: only for the last 14 days.
API /repos/:owner/:repo/releases/:release_id for getting downloads number of your assets (files attached to the release), field download_count mentioned below, but, as commented, only for the most recent 30 releases..

Update 2017

You still can use the GitHub API to get the download count for your releases (which is not exactly what was asked)
See "Get a single release", the download_count field.

There is no longer a traffic screen mentioning the number of repo clones.
Instead, you have to rely on third-party services like:

GitItBack (at www.netguru.co/gititback), but even that does not include the number of clones.
githubstats0, mentioned below by Aveek Saha.
~~www.somsubhra.com/github-release-stats~~ (web archive), mentioned below.
For instance, here is the number for the latest git for Windows release

Update August 2014

GitHub also proposes the number of clones for repo in its Traffic Graph:
See "Clone Graphs"

Update October 2013

As mentioned below by andyberry88, and as I detailed last July, GitHub now proposes releases (see its API), which has a download_count field.

Michele Milidoni, in his (upvoted) answer, does use that field in his python script.
(very small extract)

c.setopt(c.URL, 'https://api.github.com/repos/' + full_name + '/releases')
for p in myobj:
    if "assets" in p:
        for asset in p['assets']:
            print (asset['name'] + ": " + str(asset['download_count']) +
                   " downloads")

Original answer (December 2010)

I am not sure you can see that information (if it is recorded at all), because I don't see it in the GitHub Repository API:

$ curl http://github.com/api/v2/yaml/repos/show/schacon/grit
---
repository:
  :name: grit
  :owner: schacon
  :source: mojombo/grit # The original repo at top of the pyramid
  :parent: defunkt/grit # This repo's direct parent
  :description: Grit is a Ruby library for extracting information from a
  git repository in an object oriented manner - this fork tries to
  intergrate as much pure-ruby functionality as possible
  :forks: 4
  :watchers: 67
  :private: false
  :url: http://github.com/schacon/grit
  :fork: true
  :homepage: http://grit.rubyforge.org/
  :has_wiki: true
  :has_issues: false
  :has_downloads: true

You can only see if it has downloads or not.

Adam Jagosz reports in the comments:

I got it to work with
 curl -H "Accept: application/vnd.github.v3+json"   https://api.github.com/repos/:user/:repo/releases
A couple of things that I had wrong:

I needed an actual Github release (not just git tag, even though Github does display those under releases, ugh).

And the release needs an asset file other than the zipped source that is added automatically in order to get the download count.