I have a new very fast internet, however I'm not sure if PS3 supports it (my laptop for example is only able to use 1/3 of the speed). How to check what speed PS3 can use?
On your XMB, go to Settings and choose Network Settings. One of the options should be Test Connection Settings. Wait for the test to end completely, the bottom two lines should show your tested download and upload speeds
It's highly likely that every device on your network is capable of speeds that are faster than your internet connection. The real bottleneck is whatever server on the internet you're trying to talk to is capable of, and how much "real" bandwidth you have available.
Most devices on your home network are capable of at least 100 Megabit per second (wired) or 54 Megabit per second (wireless, assuming 802.11g). Most broadband connections top out at a fraction of this; the fastest available in my area is 10 Megabit per second. (Other countries vary)
However, the speed of your network devices and internet connection are only part of what factors into overall speed. When you download something from the internet, you're sending tons of data back and forth to a remote server somewhere on the internet. That server may have other clients and limited bandwidth itself. Therefore, there's a strong possiblity that you won't be able to get a full 10 megabits per second. I tend to get pretty good download speeds from Steam, and major companies like Microsoft or Google.
Layered on top of this is the fact that often your internet service provider will advertise one speed and not give you that full speed all the time (or ever!)
Note that "megabits" are generally used to give network speeds, but you may be looking at "megabytes" per seconds in your download rate. If you need to convert megabits per second to megabytes per second, divide by 8. Therefore, 10 megabit per second internet service would allow you to download (at maximum, under ideal conditions) 1.25 megabytes per second.
To summarize, there are three speeds you have to be concerned with when trying to determine how fast you can download something from the internet:
The slowest of these three determines the overall download speed.
The most real-world test for download speed on your PS3 would be to try to download something off of PSN. Any other test is likely to give you artificial results.