It's pretty easy to recognize a terrible cover letter within the first 2 seconds of reading one. People tend to forget that this document is a sales tool - you use it to sell yourself to a prospective employer. That being said, it's very easy to ruin your potential sale with just a few simple words.
The most important thing you need to do when writing your cover letter is remember that the person reading it cares about what you have to offer them, not about who you are in general. When you start off with My name is... a hiring manager is immediately going to think that he or she is about to read a life story, and they won't be particularly interested. While it may be anything but a mini-autobiography, it doesn't matter when you've already turned off the reader with those 3 words.
Starting off with My name is... is pretty irrelevant when you think about it. Your name is already at the top of the page, or it's listed as the return name in your email message; you don't need to remind them a third time. Instead, you need to focus on why you are writing this letter, and stick to just that.
It's important to keep cover letters short and very straight-forward. Hiring managers are very busy and they don't have time to read more than a few short paragraphs. Your writing needs to be engaging and interesting; you want the reader to feel compelled to read the entire thing - you don't want them to get turned off immediately. Hiring managers tend to skim through cover letters quickly, so it's important to highlight the most important details: why you are contacting them and why you are qualified. They aren't interested in much more, so make sure you keep it simple.
Your goals (in addition to eventually getting hired) are to have your resume read and to be called in for an interview, so try to use all the tools you can to make that possible. Remember these tips when writing your cover letter, and I guarantee you'll find more success in getting called for an interview.