I preach against this once a month, yet I continuously see businesses doing this. It's a bad thing.
What do I mean by account linking? I'm mostly talking about businesses that push their Facebook posts to Twitter. But there are other options, Tumblr to Twitter, Twitter to Facebook (Selective Tweets isn't as bad), or Foursquare to Twitter. Really, anytime you automatically push something from one account to another, it's going to cause trouble eventually. I'm going to base all of these problems on pushing from Facebook to Twitter though, because that's the mistake I see most often.
When you write a message on Facebook or Twitter, you are targeting a different audience. You should be thinking about it differently. First, onFacebook you no longer have to limit your message size to 14o characters. Additionally, people can comment directly on the update. This lets you potentially start a conversation that is easy to follow, perhaps drawing more people in. Either way, it's a different medium than Twitter. Twitter is 140 characters. That's it. You can have great conversations on Twitter, but it's difficult for others to follow, especially if they don't follow the person your talking to, a very likely scenario as a business account.
One other reason this is a bad idea, especially in the case of Facebook and Twitter. Facebook's Twitter integration doesn't just post the text of your status update to Twitter, it posts a link back to your Facebook account. This might sound not sound so bad, but remember, you only have 140 characters to play with on Twitter. What happens if your status update + the link go over that limit. Your message is truncated and your Twitter followers can't see the whole thing. This is fine if they are on a desktop with access to Facebook, they just click the link, they might still have to sign on, but it's not that frustrating. This changes on a mobile device. Since most twitter client's internal browsers don't save cookies, if a user clicks on your fb.me link, they have to sign in to Facebook just to see the extra charecters, guess how many of them end up reading your message?
The final reason, it gets hard to manage. Say for instance you've tasked Facebook with cross posting your status updates to Twitter, Google Plus, and a fourth specialized social media site. Then one day you start using a blogging software. When you are setting this up you are asked to link it to Twitter and Facebook to promote your posts. Sounds like a great idea, expect for one small problem. Everytime you post to your blog, your followers get hit with two messages at the same time. The original message your blogging system created and the cross post Facebook has sent on your behalf. This isn't a good content strategy for building loyalty.
There are additional reasons to avoid this. You're probably thinking, but I run all the aspects of my business, including it's social media efforts. I don't have time to post in two different places. Understandable. That's why I recommend using Hootsuite or Seesmic. They allow you to post to all account simultaneously. This lets you craft your post for the simplest network and use that elsewhere. It's not as good as crafting your message for each site, but it's a lot better than using the sites built-in cross posting functionality.