Update:
Running sudo rm /var/lib/apt/lists/partial/* temporarily allows sudo apt-get update to work.
Original:
I've seen answers to similar questions like this one, but the answers have been 'your version of ubuntu is no longer supported'.
We are using 12.04 Server which is an LTS release. As far as I know, that means precise packages should exist until support is over.
The errors only started showing up about a week or two ago. If it helps, the servers are on Amazon EC2.
I run sudo apt-get update and get the following errors:
Hit http://repos.sensuapp.org sensu Release.gpg
Hit http://repos.sensuapp.org sensu Release
Hit http://security.ubuntu.com precise-security Release.gpg
Hit http://packages.treasure-data.com precise Release.gpg
Hit http://us-east-1.ec2.archive.ubuntu.com precise Release.gpg
Hit http://packages.treasure-data.com precise Release
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates Release.gpg
Hit http://security.ubuntu.com precise-security Release
Hit http://repos.sensuapp.org sensu/main amd64 Packages
Hit http://packages.treasure-data.com precise/contrib amd64 Packages
Hit http://packages.treasure-data.com precise/contrib i386 Packages
Hit http://us-east-1.ec2.archive.ubuntu.com precise Release
Ign http://packages.treasure-data.com precise/contrib TranslationIndex
Hit http://repos.sensuapp.org sensu/main i386 Packages
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates Release
Ign http://repos.sensuapp.org sensu/main TranslationIndex
Hit http://security.ubuntu.com precise-security/main Sources
Hit http://security.ubuntu.com precise-security/universe Sources
Hit http://us-east-1.ec2.archive.ubuntu.com precise/main Sources
Hit http://security.ubuntu.com precise-security/main amd64 Packages
Ign http://packages.treasure-data.com precise/contrib Translation-en_US
Ign http://packages.treasure-data.com precise/contrib Translation-en
Hit http://security.ubuntu.com precise-security/universe amd64 Packages
Hit http://ppa.launchpad.net precise Release.gpg
Hit http://security.ubuntu.com precise-security/main i386 Packages
Hit http://security.ubuntu.com precise-security/universe i386 Packages
Hit http://us-east-1.ec2.archive.ubuntu.com precise/main i386 Packages
Hit http://security.ubuntu.com precise-security/main TranslationIndex
Ign http://repos.sensuapp.org sensu/main Translation-en_US
Hit http://security.ubuntu.com precise-security/universe TranslationIndex
Ign http://repos.sensuapp.org sensu/main Translation-en
Hit http://ppa.launchpad.net precise Release.gpg
Hit http://us-east-1.ec2.archive.ubuntu.com precise/main TranslationIndex
Hit http://security.ubuntu.com precise-security/main Translation-en
Hit http://us-east-1.ec2.archive.ubuntu.com precise/universe TranslationIndex
Hit http://security.ubuntu.com precise-security/universe Translation-en
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates/main Sources
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates/universe Sources
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates/main amd64 Packages
Hit http://ppa.launchpad.net precise Release
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates/universe amd64 Packages
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates/main i386 Packages
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates/universe i386 Packages
Hit http://ppa.launchpad.net precise Release
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates/main TranslationIndex
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates/universe TranslationIndex
Hit http://ppa.launchpad.net precise/main amd64 Packages
Hit http://us-east-1.ec2.archive.ubuntu.com precise/main Translation-en
Hit http://us-east-1.ec2.archive.ubuntu.com precise/universe Translation-en
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates/main Translation-en
Hit http://us-east-1.ec2.archive.ubuntu.com precise-updates/universe Translation-en
Hit http://ppa.launchpad.net precise/main i386 Packages
Err http://us-east-1.ec2.archive.ubuntu.com precise/universe Sources
404 Not Found [IP: 91.189.91.24 80]
Hit http://ppa.launchpad.net precise/main TranslationIndex
Err http://us-east-1.ec2.archive.ubuntu.com precise/main amd64 Packages
404 Not Found [IP: 91.189.91.24 80]
Err http://us-east-1.ec2.archive.ubuntu.com precise/universe amd64 Packages
404 Not Found [IP: 91.189.91.24 80]
Err http://us-east-1.ec2.archive.ubuntu.com precise/universe i386 Packages
404 Not Found [IP: 91.189.91.24 80]
Hit http://ppa.launchpad.net precise/main amd64 Packages
Hit http://ppa.launchpad.net precise/main i386 Packages
Hit http://ppa.launchpad.net precise/main TranslationIndex
Ign http://ppa.launchpad.net precise/main Translation-en
Ign http://ppa.launchpad.net precise/main Translation-en
W: Failed to fetch http://us-east-1.ec2.archive.ubuntu.com/ubuntu/dists/precise/universe/source/Sources 404 Not Found [IP: 91.189.91.24 80]
W: Failed to fetch http://us-east-1.ec2.archive.ubuntu.com/ubuntu/dists/precise/main/binary-amd64/Packages 404 Not Found [IP: 91.189.91.24 80]
W: Failed to fetch http://us-east-1.ec2.archive.ubuntu.com/ubuntu/dists/precise/universe/binary-amd64/Packages 404 Not Found [IP: 91.189.91.24 80]
W: Failed to fetch http://us-east-1.ec2.archive.ubuntu.com/ubuntu/dists/precise/universe/binary-i386/Packages 404 Not Found [IP: 91.189.91.24 80]
E: Some index files failed to download. They have been ignored, or old ones used instead.
Best Answer
Found this, which might be what you experience, AWS forums.
It seems switching the mirror you use to use
should do the trick.