The British nobility (still, in fact) uses a system of male primogeniture; most of the holdings of the father, including the family estate and the titles, pass on to the eldest son, while younger children receive a smaller share. Titles do not pass on to daughters, except in the absence of sons; daughters, however, can of course marry into titled nobility and thus gain one by right of her husband.

What customarily happens in the English royal family is that titles are created for all sons. The Prince of Wales is traditionally the heir apparent, while the younger sons hold duchies: the second son of King Henry IV was thus Thomas, 1st Duke of Clarence, third son John, Duke of Bedford, and fourth son Humprey, Duke of Gloucester. These are some of the traditional titles given to close relatives of the reigning monarch; they would ordinarily pass down in the male line until it becomes extinct, after which the title could then be again bestowed on a new line.

For the other sons of the royal dukes, titles may be created by the king at some point, much like for other younger sons of the high nobility: this could happen, for example, as a result of political or military service. This may be more likely than for an average member of the nobility; royal blood was obviously valued. This blood, and thus the status from being related to royalty, would dilute through the generations, so yes - eventually any advantage would be lost.

So to give an example; if a fictional king has two sons, one will be the heir apparent, Prince of Wales, and the other will be created Duke of Clarence. The Duke of Clarence has two sons: the elder will be the 2nd Duke of Clarence, the younger will have no title but one may be created for him, say Earl of Buckingham. He, again, has two sons. The elder son will be the 2nd Earl of Buckingham, the younger will have no title.

At this point it is less likely that the younger son will gain a title: he will be three generations removed from royal blood and will probably have less to do with the court and politics within it. If one continues this for several generations the younger sons of younger sons will be of increasingly low rank, though this may be corrected by a fresh marriage into the high nobility or extraordinary personal ability, leading to contributions in military or political affairs and thus fresh titles.

It is also notable that especially in pre-modern history these situations may not happen as frequently as one might expect. Infant mortality was high, and not all royal brothers had children; out of the duchies Henry IV created for his sons, none passed on, as none of the three dukes had legitimate issue (though they did have bastard sons). The younger sons joining the ecclesiastical orders was also a way in which splitting the inheritance could be avoided.

So - yes, after some generations the younger sons will lose the advantages of royal status. This is also illustrated by contemporary etiquette: a sovereign's great-grandchildren are addressed as Lords or Ladies, but thereafter there is no set form of address or title.

The current system of royal titles dates from 1917, when in the midst of World War One, King George V issued letters patent declaring exactly who would be eligible for the title Prince or Princess and the corresponding HRH. This stated that the only people automatically granted royal status are the sovereign, his or her children, his or her grandchildren (in the male line. This means that while Prince Andrew's children were automatically Princesses, Princess Anne's were granted no royal title) and the eldest son of the eldest son of the Prince of Wales. Before this, all great-grandchildren (once again in the male line) were automatically given the title Prince or Princess, with the slightly lesser designation HH (Highness vs Royal Highness). This meant that George VI, born as great-grandson of the Queen, was originally styled His Highness Prince Albert of York before being elevated to the Royal Highness level. These letters have been modified a few times. Most recently, when Prince William and Kate were expecting, the Queen issued Letters Patent to say that their child would be styled a Royal Highness, whether or not it was a boy or a girl. Sorry for any formatting issues, I'm on my phone Further explanation here