Will experience in related fields help me get into my "dream" graduate program as a non-traditional student with some failures in my past?
When I started my PhD (R1 school in engineering), I shared my office with an older gentleman who was 12 years older than me, had an average GPA, been a structural engineer for 10-15 years or so, married with kids and extensive industry experience. He applied for 3 years in a row to the same university and only that university because he lived in that town and did not want to relocate (his kids were at school, his wife had a job too) and every time he would get rejected. At his fourth trail, he was accepted. He said that he was going to keep applying as much as needed! As far as I know, nothing has changed in his application from the third to the fourth year. The professor who agreed to support him has just got a project and needed somebody with technical experience (it was construction/material related). 

I would say that getting to know the right people does not hurt! Maybe visit the department, introduce yourself and have short meeting with some professors. I noticed that many professors hold much respect for established industry people who want to go back to school. Although some feel it maybe a little risky as well. But, you never know if you do not ask or try. I do not think it is impossible, but maybe you would need to do some extra work to convince/explain your story/timing for such a decision.

I can also think of a scenario were if you work at a company that has an R&amp;D department, maybe a collaboration between the company and school can open up a spot for you in a program. In this scenario, the company can provide the school with a proposal, pay fees and (raw) materials and the school can allocate the lab, equipment and an advisor to supervise you. You can work on such a project and get a degree at the end of the day. However, I'm not sure if this applicable to your case or if this something that happens often.