I will address this simply from the standpoint of increasing the amount of time you can run, but I do want to point out that there are other factors that may influence this, such as your nutrition, hydration, other workouts around the run, things of that nature.
You are running approximately 30 minutes, 3x a week, for around 9-10 miles total distance. If you keep running at that same pace, same distance all the time, then your body cannot adapt to longer distances, and thus when you nearly double your distance, you fatigue. This is not surprising, since you are not stretching yourself. If you want to run longer than 5 miles, then you need to work on that day to day.
I personally think that you get much more out of your running if you are doing it more often, so in addition to the 3x per week, I would add in 2 days of 15 minute runs. Keep this up for a few weeks to get used to it, then start adding some time to all your runs. Don't add more than a few minutes to each run, and then give it a couple weeks to adapt to the new level.
After a few months, your daily longer runs will be in the 45 minute range, which will put you in the area of 5 miles at your current pace, and that should result in being able to run 7-9 miles for a long run.
To adapt to run longer distances, you have to practice longer distances. Right now, you are not pushing past that limitation, so it's not going to naturally get longer. You can level off wherever you want for your own personal fitness goals.
Edited to add: You are 4.0 player, which is getting pretty decent. I would assume that you have a coach, but even if not, you have to also consider if simply running more will give you the fitness you desire. If you edit your question to let us know what kind of stamina you are talking about you may get better responses. I can run 13 miles at any time due to my training, but I couldn't do the lateral movements required in tennis for more than probably a set before I'd be dragging. Where do you feel your fitness is failing you?