First, mountains are the results of the clash of the crustal plates. Without seeing the larger world picture, what I see looks possible but not likely. That doesn't rule out this map. Earth is full of unlikely formations.
@L.Dutch mentioned, rivers begin at mountains and very, very rarely cut through them. Water flows downhill. The steeper the incline the straighter the river.
However, what caught my attention first was the central valley. that area would likely be dry planes or desert. The air loses its pressure as it rises up a mountain and can't hold as much moisture. Thus, it rains on the windward side of the mountain. The air on the leeward side is much dryer. If the area between the West and the East ranges was lower than the land to the north, you might have rivers feeding a lake/marsh area there but the water has to come from somewhere.
If the winds were from East to West and the Eastern range is fairly short, you could probably get this effect but you would likely lose the green bit on the West coast where the two Western ranges meet.
If you want realism:
- Make mountain ranges
- Determine how old the ranges are. Newer ranges are sharper and older
ones rounder (look at the US Western and Eastern mountains)
- Determine how they affect the prevailing winds
- Decide where the rain falls
- Decide where the rain goes (usually to an ocean unless it goes to a
flat spot where it evaporates)
- Lay out vegetation based on availability of water and soil. Soil
is in flat spots and all over older mountains.