If you are on a perfectly clean channel, with a signal strength (RSSI) of between -20dBm and -60dBm, and you have a well-optimized TCP/IP stack and application, and high-quality 802.11g chipsets, you should be able to see as much as 25 megabits/sec (30 if both ends support frame bursting).
Note that 1 meter away may be too close. At such close range, it's possible to have a signal strength above -20dBm, which could be "too hot" of a signal and overloads the receiver. High-quality chipsets might be able to handle signals as hot as 0dBm and still receive at maximum data rates, but I've seen plenty of lesser-quality chipsets that lost their top data rates at -20dBm. 2-3m away is a better choice for top data rates.
Here in almost 2012, finding high quality G gear is pretty hard, because 802.11g is from almost a decade ago. Anyone still making G-only chipsets now or in the last 3-4 years was likely only doing it to be as cheap and small and low-power as possible (for the smartphone/tablet/netbook markets, among others), which is kind of the opposite of high quality.
The companies making high-quality 802.11 chipsets in late 2011 and early 2012 are making 3x3:3, HT40 (450 megabits/sec) 802.11n gear. And even then, they spend most of their time making sure their N rates are optimal, and less time worrying about optimizing their backwards compatibility with a/b/g.
Having a well-optimized TCP stack and an app that always keeps the TCP pipe full is good too. I recommend IPerf as a simple performance tool that knows how to use TCP effectively. If you get much better performance with IPerf than you did with the app you were running, then the app you were running is probably non-optimal. See what TCP window IPerf reports your machines are using, and make sure it meets or exceeds the "'bandwidth * delay' product" for your network (you likely need something like 20KiB or larger).
Your internet speed is independent of your Wifi strength.
Your Wifi strength may affect the internet speed you see.
Sounds contradictory but let me explain. We're going to assume for this example your internet speed is 20 Mb/s, and your Wifi link is using the 802.11g standard, which can handle up to 54 Mb/s.
The first line - Your internet speed is independent of your Wifi strength can be explained by saying that no matter what your Wifi link is, your router is obtaining that 20 Mb/s. The question only becomes, can I get that speed to my computer.
Now for the second line - Your Wifi strength may affect the internet speed you see. is because Wifi is how you're getting the information to the computer. As you move further away from the router the signal between it and your computer degrades. When you're sitting right next to it, you may get the full 54 Mb/s bandwidth. If you move down the hallway you may get 40 Mb/s. If you go down the hallway into another room with thick walls you may get 25 Mb/s. The distance and amount of obstruction can effect the signal strength you receive.
So what does this mean? It means that if you have the capacity for an internet speed of 20 Mb/s, as long as your Wifi is able to produce at least this bandwidth or above, and no one else is using the bandwidth, you'll not notice a difference in your internet speed regardless of strength. You could be at 20 Mb/s, 30 Mb/s, 40 Mb/s - it doesn't matter because your internet can only put through a maximum of 20 Mb/s.
On the other hand if your signal begins to get really poor you might drop down to having a wireless link of 11 Mb/s. Suddenly as far as you'll see, you have a speed of 11 Mb/s. The internet connect itself hasn't slowed, but the maximum amount of it that's being passed to you has been reduced. Hence your Wifi can affect the (perceived) internet speed.
TL;DR - Vast majority of the time the internet is the bottleneck in speed, so varying signal strengths for Wifi don't matter. If your Wifi link degraded so much that it was a lower speed than your internet was capable of, yes you'd notice a slow down.
Best Answer
Wi-Fi is half duplex and has more overhead than Ethernet, so you never see TCP/IPv4 thruput even as high as 80% of your physical signalling rate (known as a "PHY rate").
Plus, when sending wireless to wireless, every packet takes up channel airtime twice: once from the source to the AP, then again from the AP to the destination.
So assuming both clients are getting a PHY rate of 64Mbps to/from the AP, first we divide that in half because of wireless to wireless (=32Mbps), then we multiply by, say, 0.7 to estimate 70% efficiency (=22.4Mbps), then we divide by 8.4 to roughly convert from megabits to MebiBytes, for an estimated TCP/IPv4 throughput of 2.66 MebiBytes/sec.
Add to that the potential protocol overhead of whatever you're using to copy your files (remote filesystem protocols like SMB are often inefficient), and it's almost a surprise you're getting much more than 2 MebiBytes/sec even assuming a constant PHY rate of 64Mbps.