Can I name my website the same as someone else's if it already exists?

If I own DomainName.com but someone else with DomainName.info already has a site named "Domain Name," am I allowed to call my site "Domain Name" as well?

You are, the only drawback would be if they have the actual name TradeMarked, in which case you could still use the domain name, but you would have to differentiate your business.

Sample.

I own bigfrog.org
You own bigfrog.com

I have a trademark on Big Frog.
You can still use the domainname for bigfrog.com but have to use it for a different business, so you could name your business Big Frog Software, or Big Frog Consulting.

Also if your competition is not Facebook, CocaCola, or a big player, you are fine.
If your domain competition is in a different country, then you are golden.
And if your business is totally different, and non threatening then there is not much to worry about.