List of recent failures shows this as first build with the problem:
https://build.chromium.org/p/chromium.win/builders/Win7%20%2832%29%20Tests/builds/10981


Output from latest failed build shows:

NavigatingExtensionPopupBrowserTest.DownloadViaPost (run #1):
[ RUN      ] NavigatingExtensionPopupBrowserTest.DownloadViaPost

NavigatingExtensionPopupBrowserTest.DownloadViaPost (run #2):
[ RUN      ] NavigatingExtensionPopupBrowserTest.DownloadViaPost

NavigatingExtensionPopupBrowserTest.DownloadViaPost (run #3):
[ RUN      ] NavigatingExtensionPopupBrowserTest.DownloadViaPost

NavigatingExtensionPopupBrowserTest.DownloadViaPost (run #4):
[ RUN      ] NavigatingExtensionPopupBrowserTest.DownloadViaPost

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/1925b4789e6cea57bbd888137c202dec224bfdec

commit 1925b4789e6cea57bbd888137c202dec224bfdec
Author: finnur <finnur@chromium.org>
Date: Fri Oct 07 13:14:23 2016

Disabling NavigatingExtensionPopupBrowserTest.DownloadViaPost
on Windows due to reliable failure.

TBR=lukasza
BUG= 653856 

Review-Url: https://codereview.chromium.org/2401003002
Cr-Commit-Position: refs/heads/master@{#423844}

[modify] https://crrev.com/1925b4789e6cea57bbd888137c202dec224bfdec/chrome/browser/extensions/api/extension_action/browser_action_apitest.cc

Thank you for catching the failure and opening the bug.  Let me try bisecting as the first step.

Thank you for jumping on it. Please also monitor the progress of:
https://codereview.chromium.org/2396393002/

Interestingly I cannot repro on my Linux machine (repro = running the affected browser test).

And as finnur@ pointed out, the bot has surprisingly turned green for one build in #11011 - maybe this issue is timing-related / race-related (why else would one of the builds mysteriously succeed?).

Let me try building browser tests on Windows...

I can't repro on Windows either...

FWIW, I tried to repro failures in the other test that started failing on the same (Win7 (32) Tests) bot around the same time as the NavigatingExtensionPopupBrowserTest.DownloadViaPost test.  This other test is: PageLoadMetricsBrowserTest.IgnoreDownloads.

I can surprisingly repro a timeout in this other test, but the timeout is happening on my machine at revisions before the time when the bot turned red.  So maybe these 2 failures are bot/machine-specific (and independent / external to the Chromium's product code).  The failure mode is that test hangs when a "Save As" dialog appears (for "download-test3-attachment.gif" file).

I don't know what to do with this bug :-(

The disabled test is a regression test for a platform-agnostic issue, so I guess I don't care that much about having this test disabled on Windows.

Hmmm... this issue really seems limited to only the "Win7 (32) Tests" bot - the test seems to be passing just fine on other bots (i.e. https://build.chromium.org/p/tryserver.chromium.win/builders/win_chromium_rel_ng/builds/307944).

asanka@ suggested a plausible explanation of what is going on in  https://crbug.com/653982#c2 

I am working on a fix.

Yes, Asanka's suggestion in the linked bug resolved the issue in that case, so I expect a similar change would fix this issue as well. See https://codereview.chromium.org/2405583002 for an example fix.

The same thing started on Mac, first failing build: https://build.chromium.org/p/chromium.mac/builders/Mac10.10%20Tests/builds/8265

I'll disable the test on Mac too, to get the tree green until the proper fix lands.

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/7f2b5e5d049e63506b9c91428701d79b9057496d

commit 7f2b5e5d049e63506b9c91428701d79b9057496d
Author: treib <treib@chromium.org>
Date: Mon Oct 10 14:07:24 2016

Disable NavigatingExtensionPopupBrowserTest.DownloadViaPost on Mac
due to reliable failures without apparent cause.
It's already disabled on Win for the same reason.
See the bug for some investigation.

TBR=lukasza
BUG= 653856 

Review-Url: https://codereview.chromium.org/2404873002
Cr-Commit-Position: refs/heads/master@{#424144}

[modify] https://crrev.com/7f2b5e5d049e63506b9c91428701d79b9057496d/chrome/browser/extensions/api/extension_action/browser_action_apitest.cc

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/0afc6b3cfc1d61d2e8371b4c46cc72abd26b2a04

commit 0afc6b3cfc1d61d2e8371b4c46cc72abd26b2a04
Author: lukasza <lukasza@chromium.org>
Date: Mon Oct 10 15:44:01 2016

Using ScopedTempDir as the download directory used by the DownloadViaPost test.

BUG= 653856 

Review-Url: https://codereview.chromium.org/2407513002
Cr-Commit-Position: refs/heads/master@{#424157}

[modify] https://crrev.com/0afc6b3cfc1d61d2e8371b4c46cc72abd26b2a04/chrome/browser/extensions/api/extension_action/browser_action_apitest.cc

https://build.chromium.org/p/chromium.win/builders/Win7%20%2832%29%20Tests stays green, so marking this as fixed.