I agree with the vast majority of what IR has said so far in this thread, only thing I might counter is that not all of these jobs are necessarily extremely highly sought after, nor do they always require a wealth of experience, as long as you are willing to accept a junior role and work your way up in a company and/or industry (and face it, if you wish to do design and prototyping, this is pretty much the only way to achieve that end).
I work within and alongside the R&D group of a large, multinational company that provides structural building materials for the building industry (I'm being intentionally vague here on purpose). I'll try and summarize my thoughts regarding this concept of prototyping and 'inventing' below:
1) First, as IR has alluded to, 'inventing' something new is going to be extremely rare, most everything is an iterative process, defined by taking something that already works and making it more efficient to manufacture and/or sell.
2) Let's talk about degree requirements & education, you will need to have a baseline understand of how the physical world works and interacts, that means not only understand applied mathematics, but also have a good basis in both physics, chemistry, and depending on your field of interest, biology as well (really just applied chemistry). You get the absolute bare basics of understanding once you've received a Bachelors of Science with an Engineering Degree (Civil, Mechanical, Chemical/Environmental, etc). A BS in engineering is an absolute bare minimum, and it does not give you very much in the ways of skills to succeed, those will begin to be acquired once you actually enter the workforce. I would say that an Engineering Technology degree does not meet this requirement, and I would not recommend a Technology degree if you wish to enter R&D, unless you are most interested in technician work.
3) Is there a saleable approach for your new idea or 'invention'? Is there demand for such a thing out in the field (i.e. is this something you can sell?). If you don't have a market or customer base, you're going to have a hard time getting justification to pursue your idea. While this is a simple explanation, it needs to be kept in mind.
4) Next you need to have a basis of understanding of: is my idea buildable? (this will rely on your understanding of math, physics, chemistry, etc). Can my idea be created in an R&D setting?
5) After some very basic review to verify your idea, next step is finding time to devote to your idea. Generally speaking, you aren't going to be in a position to simply be hanging out at work and thinking up ideas to work on. So this part will be difficult. You will need funding, you will need equipment and technician help, you will need time to develop.
5a) This is where the points made above in this thread apply. Believe it or not, the engineering department for just about any company isn't swimming in cash. In fact, funding in general is extremely tight. So you will need to build a proposal justifying why time and money should be spent on your 'invention', and that will need to be presented to leadership, or whomever holds the purse strings for the company you work for. This is where your alternate humanities and technical writing classes would be handy, in that they give you some ideas how to write, approach, and talk to non-technical individuals for funding, etc.
6) By some miracle, you've come up with a good idea, you have a customer base, and you got funding from management to run a pilot/prototype line. Does your new 'invention' fit onto existing infrastructure, or do you need new equipment to manufacture? If you need to purchase new equipment, that may require multiple millions of dollars, years of building, and production area to acquire, which could be difficult to justify to management, it's always better to make something that fits within existing assets the company you work for owns.
6a) Now what? You need to prototype and troubleshoot production of the product. Things never work out perfectly, and you need to find out the product design errors that reduce yield, create unsafe maintenance or manufacturing conditions, reduce costs to manufacture, etc. Reducing costs is one of the most important parts, because if the product costs more to produce and ship than the market will bear, it's going to be dead in the water.
7) Several years later, these issues have been resolved. Now you need to address whatever regulatory hurdles that stand in the way of your product entering market. Ideally these are thought through during prototype stages, but often times each jurisdiction and/or state is unique. California is the wealthiest state in the US, and has some fierce regulatory hurdles that need addressed before entering that market. You will always want to sell your product in California, as the amount of customers and money is very high, but can you get over the hurdles required to enter that market?
8) Everything discussed above may take 5-10 years to get to completion, will require significant time and attention paid to non-technical personal. How do you communicate with these individuals? How do you get them to understand the importance or usefulness of your idea? How do you get them to provide you funding?
This is a very basic view of the process, it's not all encompassing, and it's not intended to dissuade you, just to help you realize that your idealized view of this concept may not be totally grounded in reality. If this all acceptable to you, then I would suggest spending more time talking to a college and getting a feel for their programs and how you may enter such programs (Verify the individual program is ABET accredited. Just because the college has some programs ABET accredited, doesn't necessarily mean all their programs are).
