Only the newest profile created is able to be played online. For all others the only option is "Play offline". This stops you being able to play on servers with Online Mode set to true.

For example, if I only had a profile called "1.6.1" it would let me play online. However if I then created another profile called "1.6.2" then this new one would be online, yet "1.6.1" would then become "Play offline" only.

There is a launcher/dev. console error:
"
Trying to log in...
Logging in with access token
Couldn't go online
net.minecraft.launcher.authentication.exceptions.InvalidCredentialsException: Invalid token.
at net.minecraft.launcher.authentication.yggdrasil.YggdrasilAuthenticationService.makeRequest(YggdrasilAuthenticationService.java:123)
at net.minecraft.launcher.authentication.yggdrasil.YggdrasilAuthenticationService.logInWithToken(YggdrasilAuthenticationService.java:97)
at net.minecraft.launcher.authentication.yggdrasil.YggdrasilAuthenticationService.logIn(YggdrasilAuthenticationService.java:50)
at net.minecraft.launcher.ui.sidebar.login.NotLoggedInForm$1.run(NotLoggedInForm.java:162)
at java.util.concurrent.Executors$RunnableAdapter.call(Unknown Source)
at java.util.concurrent.FutureTask$Sync.innerRun(Unknown Source)
at java.util.concurrent.FutureTask.run(Unknown Source)
at java.util.concurrent.ThreadPoolExecutor.runWorker(Unknown Source)
at java.util.concurrent.ThreadPoolExecutor$Worker.run(Unknown Source)
at java.lang.Thread.run(Unknown Source)

Going to play offline as 'chumps52'...
"

Screenshots attached. Does not matter if Minecraft is modded or not.