I think something is broken or messed up Ubuntu repo management wise. I too have the same errors when I try to update a 16.04.2 LTS system.

It appears that some of the repos pointed to in the sources files are either new or taken offline. This results in an incomplete update command execution and thus the upgrade command fails as the packages list is not updates/complete/valid.

I have tried changing the sources from the UK to US, FR, IN, GR, DE so its not a individual repo problem - it's bigger than that. But until this is fixed the systems will not update - why there maybe so many threads on cron updates not working too.

The back end of another failing "# apt-get update" command:

Code:
Err:20 http://security.ubuntu.com/ubuntu xenial-security/main Sources
  Connection failed [IP: 91.189.88.152 80]
Ign:30 http://security.ubuntu.com/ubuntu xenial-security/restricted Sources
Ign:32 http://security.ubuntu.com/ubuntu xenial-security/universe Sources
Ign:41 http://security.ubuntu.com/ubuntu xenial-security/multiverse Sources
Ign:51 http://security.ubuntu.com/ubuntu xenial-security/main amd64 Packages
Ign:53 http://security.ubuntu.com/ubuntu xenial-security/main i386 Packages
Ign:61 http://security.ubuntu.com/ubuntu xenial-security/main all Packages
Ign:69 http://security.ubuntu.com/ubuntu xenial-security/main Translation-en_GB
Ign:71 http://security.ubuntu.com/ubuntu xenial-security/main Translation-en
Ign:79 http://security.ubuntu.com/ubuntu xenial-security/restricted amd64 Packages
Ign:87 http://security.ubuntu.com/ubuntu xenial-security/restricted i386 Packages
Reading package lists... Done                         
W: The repository 'http://archive.ubuntu.com/ubuntu xenial Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: The repository 'http://archive.ubuntu.com/ubuntu xenial-updates Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: The repository 'http://archive.ubuntu.com/ubuntu xenial-backports Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: The repository 'http://archive.canonical.com/ubuntu xenial Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: The repository 'http://security.ubuntu.com/ubuntu xenial-security Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
E: Failed to fetch http://archive.ubuntu.com/ubuntu/dists/xenial/main/source/Sources  Connection failed [IP: 91.189.88.152 80]
E: Failed to fetch http://archive.ubuntu.com/ubuntu/dists/xenial-updates/main/source/Sources  Connection failed [IP: 91.189.88.152 80]
E: Failed to fetch http://archive.ubuntu.com/ubuntu/dists/xenial-backports/main/source/Sources  Connection failed [IP: 91.189.88.152 80]
E: Failed to fetch http://archive.canonical.com/ubuntu/dists/xenial/partner/source/Sources  Connection failed [IP: 91.189.92.150 80]
E: Failed to fetch http://security.ubuntu.com/ubuntu/dists/xenial-security/main/source/Sources  Connection failed [IP: 91.189.88.152 80]
E: Some index files failed to download. They have been ignored, or old ones used instead.
Grateful for any clues, but I have been playing with this problem for 4 days

Also it's not a DNS or proxy issue - I have been around those and conducted multiple look-ups from various parts of the internet.

Cheers