Of course, when this happens, there is so much plant life that you can't see anything else, certainly not fish! What can also happen is the plants stop the movement of water, causing stagnation. Eurasian Watermilfoils (Myriophyllum Spicatum) actually causes stagnation! I have actually experienced seeing this many times when cleaning ponds out. Plants are good for a pond, because they take the nitrates out of the water and help to keep the water clear, but it's a hard balance to strike.
Some Koi keepers don't put plants in their ponds - mainly because the Koi destroy them or eat them! There are ways round this, though. I have Koi in my pond (over 22 inches long) and if you get a well-established lily you have a good chance of success. However, buying small plants, lilies or marginals etc in small pots and expecting them to do well is a waste of money. You need to buy big, well-established plants - if you are lucky enough to hear that somebody is having their pond cleaned out they are the plants to go for!
Having an additional vegetable filter will help to stabilise the nitrates, in your pond if you're lucky enough to have the space and conditions for one to be built. All ponds are different - their location can have a strong effect on them, and there are many things that need to be taken into consideration, which is why we come out to see you.
For example filters: you might have a small filter recommended, which will do the job, but what you don't know is that you will have to clean it out every week, which will upset the biological side of the filter. Another thing to be taken into consideration is the fact that your fish will get bigger! On top of that, the pond being built in the wrong place (under trees), leads to leaves dropping into the pond, and is one of many reasons why regular pond cleaning and maintenance is so important.