That infuriated me last year when I finally tried to create a Twitter account.
During registration I was given a choice between providing an email address or a phone number. I provided an email address, only to immediately have my account locked until I provided a phone number. Well, I wouldn't have given my email address if I knew a phone number was required anyway! It wouldn't even let me visit my account settings to delete my account, and I never got a reply when I emailed Twitter support.
I had a similar experience. In my case they allowed my account to be opened without a phone number as I requested that they delete the account because I didn't want to provide my phone number. I deleted my account soon afterward anyway.
Be careful, my 10 year old account was permanently suspended without warning and support is no help; I suspect it's because I didn't have a phone number on my profile.
During registration I was given a choice between providing an email address or a phone number. I provided an email address, only to immediately have my account locked until I provided a phone number. Well, I wouldn't have given my email address if I knew a phone number was required anyway! It wouldn't even let me visit my account settings to delete my account, and I never got a reply when I emailed Twitter support.