The analogy that I've used on other forums when similar subjects have come up is comparing electronics to carpentry and construction.
An electrical engineer is akin to an architect - someone who (presumably) knows everything about the design and construction of buildings. Construction techniques, strength of materials, costs, analysis techniques, project management, design and aesthetics, etc.
Not having a degree and formal training in architecture doesn't mean that someone can't take up carpentry and progress in skill at their own pace, say from building a birdhouse kit, to building a shed, to creating beautiful furniture and cabinet work, to building and designing their own houses (as long as they understand the building codes).
An electrical engineer is trained in the math, physics, and "design patterns" for developing electronic devices. It is a demanding degree and, theoretically, once you get the degree you should be comfortable designing anything from an electronic toy, to a power supply, to PC boards with hundreds of surface mount components. At the advanced levels, or with post-graduate work, you would even be able to design and fabricate new integrated circuits and CPUs.
As an electronics hobbiest, you can start with electronics kits that, depending on how closely you study them, will teach you some basic fundamentals of electronics design. You can start with a small set of tools, a parts cabinet, and some good books or websites and design small, interesting, and handy gadgets. Along the way you will develop your skills in construction, debugging, and testing circuits and begin to acquire the basic math skills (V=IR, RC constants, series vs. parallel calculations, digital logic, etc.) and learn the rules of thumb of how to combine useful components and subassemblies to create more complex designs. In other words, you can progress as far as you want to up to a certain point. You aren't likely to be able to build a TV from scratch or even develop a robust commercial product without some formal training, for example. Then again, a weekend carpenter, no matter how skilled, isn't going to design a skyscraper.
As I said before, using microcontrollers like the Arduino that let you abstract a lot of complexity into software and the fact that there are many really easy to use sophisticated devices and subassemblies (accelerometers, GPS on a chip, computer interfaces, Bluetooth boards, computer interfaces, LCD module drives, etc.) means that this is actually a very exciting time to become an electronics hobbiest (or professional).
Here are some good links to things related to this rebirth of hobby electronics:
http://makezine.com"
http://sparkfun.com" - check out their cool selection of sensors, microcontroller tools, components
http://arduino.cc" - mentioned this earlier
http://ladyada.net/make/index.html" - kits, tutorials, and a great forum