Are most contractors capable of building any home?
Although David Moore is somewhat correct, there are a lot of varied explanations to your question depending on where you live, the market etc.  In large markets, there are large corporate home builders that have multiple crews covering all trades and do a turn key package. Many do custom homes from your plans. Many however will only build your home on land they have and sell you. Mega builders are not as flexible as smaller builders.

In more rural areas, or areas where development is not large, the contractor pool is much smaller. In my area around Portland Maine and most of New England, custom home builders are mostly general contractors. They have preferred sub-contractors for foundations, framing, electrical, etc, etc, etc.  There is absolutely nothing wrong with using a general contractor, in fact, they often are very competitive and build a great product.

The key to getting what you want is picking the right contractor based on research, referrals, and interviews with existing customers. Check with your local BBB, State licensing bureau or similar to see if the contractor is in good standing and financially sound.

When you sit with you contractor, the devil is in the details. Know exactly what you want. Don't fall for the "allowance" quotes.  Be specific on what type, brand, model of everything you can think of, or pick from what the contractor has to offer.  Example: lighting fixtures can be $15 or $80 each, plumbing fixtures can range from $50 for a bath faucet to over $300, counter tops go for $20/square foot to over $100/square foot. The same goes for windows, doors flooring, cabinets, counter tops, wiring devices, tile, the list goes on and on. Quality costs money, but spec grade materials can be very cheaply made.  Insist on a full check list, specify all options, materials and grade/models of appliances etc.  This is a big job, but if you want to be happy with the finish product, know what will be going into it. 

A good general contractor is going to spend the time and help educate you and assist you in making these important decisions. The more specific you are up front, the more accurate your quote will be and less $$$ overruns. Change orders are very expensive.

Always be proactive, inspect materials before they are installed. Be sure the right stuff is being used and your specifications are being followed. Having an open line of communication with the GC is the key to assure a smooth job and catch problems before they happen.   Good luck