In my profile I went to Edit Profile & Settings > Edit Email Settings
I entered a new email address I want to use to log in.
Why is it still possible to log in with the old email address?
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The email preference that you edited is not the login information that is being used to log in, but the main contact email that Stack Exchange would use to contact you (newsletters, notification digests, etc).
If you want to modify your previous login method, you need to remove it from Profile > Edit Profile & Settings > My Logins instead of Edit Email Settings.
I think you're confusing two things together:
(1) Email Settings
(2) Your Logins
Email Settings lets you set the preferred contact address. If you add additional email addresses to your logins and set StackExchange login passwords you'll have to delink those logins from your account separately.
If you mean the email preference information in Email Settings, then Dan is 100% correct.
If you mean why can you still login with your 'old' SE email/password combo, go to your My Logins page and see your logins you have authorized for your account and remove the old address. (The link to that page is also in Dan's answer). Your "old" and "new" email addresses are both seemingly tied into your SE account currently. (Just as an FYI, as a moderator I can see that information)