Weight Training and Running the Same Day

I've been weight training consistently for a couple of years now. In August I started running. I begin my run at 5:45 in the morning and typically run 4 times per week. I weight train 4 - 5 times per week and generally begin my workout around 6:00 p.m. Are there any disadvantages to either running or weight training by doing both the same day? My goal in weight training is to add muscle, and my goal in running is to increase cardiovascular health and reduce body fat. I'm 6'5', 230 pounds, 35 years old, probably around 15% body fat.

No problem, doing it the way you are.

The only issue here is that your goals are somewhat at odds with one another. For cardiovascular health per se, running for more than something like 30 minutes 3x per week is unnecessary, and can be catabolic (muscle-wasting). Running more than this may improve your running, but this is more through improvement in neuromuscular efficiency than improvement in cardiovascular capacity.