It can possibly increase your internet speeds, but think of it like this: I pay $65 a month for 25+mb/s internet speeds. That being said I almost NEVER get anywhere around 25mb/s. You are limited by the servers you are connected to and their connection speeds, among other things. I used to work for a computer repair place that paid for 1 gigabit internet speed. We practically NEVER got speeds that fast, but it was certainly much faster than the internet that I have. So as far as I can tell it scales. Basically the better your internet, the faster speeds you will get, though the MAX speed will almost never be the speeds you see, though your network will be capable of those speeds under the right conditions (for example Steam servers are VERY fast and you will see download speeds of 10's of mb's a second)
Long story short, it might help but probably not, and if so probably not by much.
If you do upgrade your network card, post speed tests before and after the upgrade.
Edit: To clarify, 10-100mb/s is INSANE download speeds. If your internet is going slow, then clearly you are not MAXING out your card's capabilities, so upgrading it probably won't help. Gigabit networking equipment is the new standard, and it certainly is better to be able to download at 1000mb\s than 100mb\s, but you probably won't see speeds close to either of those most of the time, even if you have an amazing internet package.
Edit2: The best way to improve internet speeds is to get a better internet service. The US has relatively slow, expensive internet (compared to much of asia and europe) So another thing you can do is contact your congressman/representative and DEMAND that the US stay at the forefront of internet technology.
It is still lower than you nic card speed so replacing the card will have no impact.
You have to be somewhat careful with the values you get from a speedtest site. You need to try multiple to make sure it is not the location of the testing site causing the issue. Make sure you test at different times of day to see if it makes a difference....in the middle of the night generally gives the fastest results.
You may want the ISP to come to your location and test with their equipment and see what the maximum they can get. If they too can only get 1.3m then you can't do a lot better.
Remember almost all ISP are "best effort" they will say things like "maximum" download speed. They seldom if ever promise a minimum speed.