Can you please either:
1. file a feedback report and put #wifi-issue in the comment section, or 
2. navigate to chrome://net-internals->ChromeOS->store logs and then attach the resulting log file to this bug?

Please do the above immediately after you've repro'ed the issue.

Sounds like the "Share this network with other users" button was not checked when the credentials (password) for the network was first entered? 
As a quick check, once you sign in, please click on the wifi icon and then on the network name, that should open up a settings dialog with network information about the SSID you are connected to (see attached screenshot). It should not say "Access to this network is protected", but "This network is shared with other users."

I was able to reproduce your situation this way - made my device forget all networks, by going to the same settings panel and clicking "Preferred Networks" and then removing all of those preferred network from the list. Next, I connected to a WPA2 network without checking the "share" box and then logged out. My device said I was offline on the login screen and said SSID appeared much lower than others in the wifi menu on the login screen.  

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Response to Comment 2
Unfortunately, this issue ONLY happens right after you turn on the Chromebook and once you sign in, the issue is no longer happening.  Therefore, I don't think either of the things you requested would yield any useful results.  However, if I am wrong about that, let me know.

Response to Comment 3
Interesting that you were able to reproduce and possibly resolve this issue on your Chromebook.  However, it seems that you are probably using a Chromebook in "standard" mode, while we have ours connected in a "domain" situation by enrolling them using the "Enterprise Enrollment" option and managing them using the Chrome Management Console.

+ Steven for networking UI

Re#c6: Enterprise enrolled or not should not have anything to do with this. 
Like I mentioned in #c3, please check that your network show "This network is shared with other users." 

Please clarify what UI area you are describing, specifically:

(3) Where is 'the Wi-Fi icon shows an "x" instead of the expected bars'?
(4) Where is 'clicking on "no network"'?
(4) Where is '"WN-Chromebooks" is NOT at the top'?

Also: What if any network icon is present in the bottom-right corner of the screen?

This was reported on M-55, is it possible to report whether it occurs in M-56? The system tray ("quick menu") changed signficantly in M-56.

re#C7
After signing in, "This network is shared with other users." IS there.  See attached screen shot.

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	Screenshot 2017-02-08 at 1.52.59 PM.png 28.8 KB View Download

Networks connected to in the login screen are always shared, I don't think that is the issue here.

I am much more interested to the questions in comment #8.

re#C8
(3) Where is 'the Wi-Fi icon shows an "x" instead of the expected bars'?
Answer: I am referring to the Chromebook "network" icon located to the left of the clock. In this case, it is showing "X" instead of bars when it IS connected to Wi-Fi right after turning on Chromebooks

(4) Where is 'clicking on "no network"'?
Answer: after clicking once on the bottom-right/clock area, approximately the 4th choice down is SUPPOSED TO either show "Connected to WN-Chromebooks" (when connected to Wi-Fi) or "no network" (when NOT connected to Wi-Fi). In this case, it is showing "no network" when it iS connected to the Wi-Fi

(4) Where is '"WN-Chromebooks" is NOT at the top'?
Answer: when you click on the choice mentioned in the previous answer, and get a list of SSIDs, the one you are connected to is SUPPOSED TO be at the top of the list (in bold font).  In this case, even though you really ARE connected to the Wi-Fi, NO SSID is shown in the list as being connected to (at the top in bold) and the one that is IS connected to is not in the list at all (at the top or anywhere else)

Also: What if any network icon is present in the bottom-right corner of the screen?
Answer: it shows an "x" even though you ARE connected to the Wi-Fi (see the first answer above /\)

Thank you for the answers! Follow up questions:

1) Can you reproduce this in 56? (The 'no network' icon will look different, but otherwise I the behavior should be the same, unless the problem has been fixed).

2) Could you file a feedback report after this occurs, and include "stevenjb@google.com" in the summary? The feedback report will include logs that may help us identify the root cause.

cernekee@ - I can not think of any reason that the Chrome code would be the cause here. We limit the number of networks to the first 100, but the connected network should always be the first network returned by shill::kServiceCompleteListProperty.
https://cs.chromium.org/chromium/src/chromeos/network/shill_property_handler.cc?q=shill::kServiceCompleteListProperty+package:%5Echromium$&dr=C&l=412

Sorry for the delay...

re#C12
1) yes, this is still happening in 56 - but as you mentioned the "no network" icon now looks different -  blank instead of having an "x" in it

On a related topic, I just noticed that this issue is not happening AT ALL on an older model Chromebook that we have: "Samsung Series 3 XE303C12-A01US".  I am not sure if this model Chromebook has ever experienced this issue - and of course I have no idea why this type of Chromebook is behaving differently - but it could be a clue as to what is going on.  There must be something different about this model Chromebook?

2) I just filed the feedback report that was requested.  However, since the issue only happens before you sign in, I don't know if there will be anything helpful in that report.