Lol, not my falt that screen shots are too big
Yes it is. I can fit a 1280x1200 screenshot into 47kb. I could probably fit it into less. There are a lot of ways you can shave off excess file size.
1. Make it smaller. My 1280x1200 screenshot would work just as well as a 800x600 screenshot, although it starts to become more difficult to read text. In game screenshots though, there is little harm in making the screenshot 800x600, even 640x480. A good program for resizing images is
Microsoft Office Picture Manager. If you don't have a copy of Office, you can try using GIMP or Photoshop.
2. Crop it. You can even do this in Paint and still get excellent results. The idea behind cropping is to take out the bit of an image you don't want to show. For example, if I were to post a picture of someone holding a Dr. Pepper, and I only wanted to show the can, I would crop out the person, the background, and everything else besides the can. Use google to find a tutorial for cropping using your favorite image editor.
3. Don't save as a .BMP file. They are not compressed. Export it in either .PNG or .JPG formats. You can use .GIF format too, but they can only be exported in indexed color mode. See number 4 for more details.
4. Export your picture using indexed colors rather than RGB. Pictures are usually exported in 24-bit color mode. Indexed mode is 8-bit color or 256 colors. You may recall running programs in 256 color mode before, and noticed how it distorts your desktop. However, you can tweak the color pallet, so if you take a picture of a blueberry, the pallet will use 200 shades of blue, and not 8 shades of blue because it's pallet also contains 8 shades of red that you aren't using.
You can also force even lower color modes, such as 6-bit (64 colors) and 3-bit (8 colors). Again, you can tweak the color set (some programs will do it automatically for you) and you can sometimes get decent quality pictures with very few colors.
Here is a lovely comparison for your viewing pleasure.
24-bit (13.6kb)

8-bit (5.70kb)

6-bit (3.91kb)

3-bit (2.18kb)

Here is a graph showing the relationship between the pallet size and the image size. The Y axis is the size in kb, the X axis is the number of colors. The X axis is a logarithmic scale, so the value 7.224719896 should be read as 10
7.224719896 or 16777216.00248981664792046490
4922 colors. It's actually 16777216, the crazy decimal places are caused by rounding errors.

This graph also shows proper cropping, resizing, and color optimization, at 3.88kb.
Have fun and stop trying to upload 5MB images.
EDIT: Made graph even smaller, I used an impossible color pallet previously, so the program reverted it to RGB mode and bumped the file size back to 97kb.