I prefer to use getters rather than constants. Those getters might return constant values, e.g. public int getMaxConnections() {return 10;}, but anything that needs the constant will go through a getter.
One benefit is that if your program outgrows the constant--you find that it needs to be configurable--you can just change how the getter returns the constant.
The other benefit is that in order to modify the constant you don't have to recompile everything that uses it. When you reference a static final field, the value of that constant is compiled into any bytecode that references it.
