The answer of course, is YES you can. It is no longer required to buy a complete system from a solar company, although you can if you want to save time, nowdays you can build your own panels and hook them up yourself. Buying components from a solar supplier is an option but will cost more than a DIY Solar Cell system.
It does require some research to learn how to do this effectively. But most handyman types can learn it easily and the learning curve is fast. There are a number of places you can find both free and paid manuals on how to do this. Plenty of government websites do exist as well, that can provide you lots of great information the government has already found helpful in the construction of systems.
It is always a good idea if you plan to "tie in" to the local electric grid, in order to sell your excess power, to first check with the electric company. Their rules are VERY strict about the system setup and how to do this. You can not just "plug in" with solar or wind generator systems! There is also usually a fee they charge to come out and inspect your system and do the "tie in".
Many times a home can use more power than a small system can generate. Realistically, unless you have a small energy-efficient home already, then do not expect to completely eliminate your dependence on the electric company. At best you can hope that most of your power can come from your own system, with a minimal dependence on the power company only during peak periods (summer heat). Still this is not only going to save you a ton of money, but also is much better for the environment!
The overall cost of the system really must be looked at before you begin to build. You will need multiple solar panels, not just one. Plus the wiring, batteries, etc. After figuring out the overall cost, figure out how much your monthly bill is, and what portion of that you think you can get rid of. Look at your electric bill and see how many Kilowatt hours (KWh) you currently use, and the cost for those KWh. You will want to know how long it will take to "pay back" the cost of your system. If you build a system at home using free or cheap materials (can you say LAND FILL or CONSTRUCTION TRASH?) Then it should be a short time to pay the system back.
Have people really done this? Absolutely. More and more people every day are standing up against foreign oil dependence as a whole and deciding to make their own biofuels, biodiesel, and now electricity. Just a few short years ago the only way to get solar power was to buy it from a company that manufactured the panels. But now enough people have built their own, that it is possible to do it yourself too. Solar Cells are not rocket science. Especially when you have guidance from someone who has done it before!