One of the dumbest myths I've encountered is that applying nitrogen fertilizer to your fields will kill your clovers, which fix their own nitrogen, and so make you need to apply ever more nitrogen to stay even.
Nitrogen does not kill clovers, but it does make life easier for other pasture species that do not fix their own nitrogen, and so makes them more competitive with clovers. If the pasture is not well managed and those other species, often grasses, are allowed to grow tall and rank they will shade out the shorter clovers. Clovers do need fertilizer but it is phosphorous and potassium that makes them thrive. Other species need these nutrients too so it's always good to correct for deficiencies.
Clovers can give you some good visual indication of soil health without chemical testing. If you have dark green thriving clover stands among stunted yellow grasses then the soil is likely to be nitrogen deficient. In some cases the very existence of clovers is an indication of nitrogen deficiency. If clovers are small or difficult to establish while grasses are doing well then the soil PH is likely lower than clovers prefer. Raising it with lime or bio-char is smart even though the grasses seem happy since it will improve soil chemistry so that existing soil nutrients are better utilized. The effect of raising PH can be similar to applying fertilizer though what it actually does is make existing nutrients more available to the plants.
Clovers can use some nitrogen when they are young to get established, and they prefer to get their nitrogen directly from the soil rather than fixing their own. It is actually rhizobial bacteria that nodulate on their roots that do the nitrogen fixing, but they don't do it for free. They in effect charge the plant, trading nitrates for plant sugars. If clovers can get nitrogen for free then they will keep their sugars for their own use and grow better as a result. This is fine so long as the nitrogen doesn't run out, but if it does there may not be any surviving rhizobia to trade with, leaving the clovers unable to fend for themselves in the wild so to speak. Rhizobia can be directly affected by applied nitrogen since it lowers PH. When PH is already low it can be detrimental to them. This is yet another reason to raise PH toward neutral. Lime and or bio-char seem to stimulate rhizobia (and AMF too!).
The decision for a pasture manager about applying nitrogen or trying to get more nitrogen from clovers is mostly an economic decision. Nitrogen is expensive and getting more so as stocks of natural gas dwindle or are in demand for other uses such as home heating and industrial energy.
Increasing the percentage of clovers in pastures can increase nitrogen but they aren't free either. Clover seed is expensive and often does not persist from year to year. There is labor involved in seeding too. Clover isn't as productive of dry matter as grasses so there is a drop in total pasture productivity as clover percentage increases. But, clovers have high protein levels and improve the quality of some pastures.
Usually total production drops as clover percentage increases and so stocking rates must be lowered. This may be a good trade off if land is cheap. It takes more land to handle an equivalent stocking rate. But this is often not the case when everything from taxes to fencing to water systems to labor and fuel are considered.
There's no pat answer. Each manager needs to sharpen his pencil, get accurate information and do cost benefit analysis. There are many variables to consider, even some odd ones such as the moderating effect clovers can have when animals are grazing endophyte infected fescue. The toxic effects are lowered, which can increase animal performance. If nothing else their consumption might rise since clovers are very palatable - pasture candy - while fescues can be bitter and coarse.