^^^I don't know that I have ever run more than 20 miles in a week, but I can safely do speedwork and/or run hills on trails whenever I feel like it (1-4 times a week). If I wanted to be, I could quickly be ready for a wide range of race distances (no desire). A lot depends on a person's goal(s), other training, and previous experiences. Cycling, hiking, weights, and plyometric exercises work well for me as cross training.
I'll build on some of the concepts in this thread with some additional tips for beginning/intermediate runners: If you aren't in an important race, walk down the steep hills. Learn about proper form and cadence. Pay attention to technique/form and how you feel during and after a run. If you feel an injury, stop running that day and monitor it/treat it.. If your form is falling apart and you begin to stomp, slouch, or feel signs of overused hips, knees, shins, or ankles, take walking breaks or call it a day. If running is new to you or you are out of shape, don't expect to be ready to go fast or far until you have spent time adapting to running and developing good form. There are lots of good articles on these subjects on the internet (Active.com, Runner's World etc.), as well as good training plans for various distances (5K, 10K, half-marathon). Learn about technique drills and strength/power exercises that can increase agility, improve technique, and reduce risk of injury (plyometrics are an example). Learn stretches to do after your runs, especially focusing on problem areas. Invest in foam rollers or tennis balls to massage problem areas (or pay for sports massages). Avoid running on hard surfaces, especially concrete. Adapting to blacktop and even some tracks may take time. Trails, grass, and dirt are great, but be careful of uneven spots that could lead to ankle sprains and other injuries. Consider cross-training, such as cycling, stair climbing, weight training, plyometrics, strenuous hiking/climbing, swimming, yoga, skiing, etc. to stay fit & increase strength without overdoing the running. Carefully trying new things and mixing up stale routines will generally increase overall strength and fitness (especially if you have stopped improving).
Don't expect to improve much at any of the above activities if you don't exercise at least 20-60 minutes 3-6 days a week, with intensity and strength training included some days. Interval training builds speed and strength. A longer session of exercise about once a week helps endurance, but a long hike counts if fitness for hiking is your goal. With persistence, you can become very fit for intense hiking at the middle ranges of the spectrum (of time spent) and can even become a decent middle distance runner (1-10 miles). Depending on training intensity, don't forget to take it really easy for about a week every 6-8 weeks or so, as well as after unusually long and intense exercise/races..
Some statements in the thread have been geared more to competitive half-marathon, marathon and ultra distance running, which is only worth carefully undertaking if you really want to do it. For most people, running beyond about 15 miles (or even beyond 3-6 intense miles) has very little added benefit and increases the odds of injury due to bad form/technique and/or overuse. Diminishing returns and increased injury risk come into play at some point in your mileage and intensity level for a day, week, or training cycle. That applies to all types of exercise. If you love ultra endurance exercise, I'm sure you figure out ways to make it work and some are genetically fortunate in that realm.