It doesn’t interfere with the network directly, but as we are dealing with a µC with limited resources and no OS as you would find on a computer, memory management is something the programmer has to take into consideration.
String objects make use of dynamic (heap) memory and if they grow out of the pre-allocated space (default 16byte) they might get relocated, leaving a gap in their previous spot. Over time these gaps might get so many but so small that other objects (including the system) who might acquire dynamic memory won’t find a free spot big enough for their data and fail to work. This is when the ever growing heap fragmentation causes trouble.
Full fledged OSs regularly perform garbage collection and defragmentation, to “compress” the data to get the memory back, but no such thing is happening here.