According to this Wikipedia article, the number of Representatives was limited to 435 in 1911 (bumped to 437 when Alaska and Hawaii were admitted and then bumped back to 435 four years later).
How (and why) was the decision made to stick with 435?
Pretty much correct Wladmir, this article shows the growth over time, but it was all due to the Permanent Apportionment Act of 1929: http://www.thirty-thousand.org/pages/Why_435.htm
As noted here: http://artandhistory.house.gov/highlights.aspx?action=view&intID=200 this came down to a fight after the 1920 Census, where apportionment did not happen, so it was mandated to be at the current level which was 435 at the time.
A battle erupted between rural and urban factions, causing the House (for the only time in its history) to fail to reapportion itself following the 1920 Census. Signed into law on June 18, 1929, the Permanent Apportionment Act capped House Membership at the level established after the 1910 Census and created a procedure for automatically reapportioning House seats after every decennial census.
At some point it had to stop, although it seems as if Congress could change it again if they wanted to, so long as the apportionment stay as constitutionally derived.
Just to expand on MichaelF's answer, the House began to debate the increasing size of Congress after the 1910 census. There was worry that the House was becoming unwieldy, but as Representative Edgar Crumpacker stated:
Members are . . . supposed to reflect the opinion and to stand for the wishes of their constituents.
If we make the ratio [of persons per Representative] too large the idea of representation becomes attenuated and less definite. The personal interest of the voter in his representative becomes less important to him, and we may lose something of the vital strength of our representative form of government.
So there was concern on both sides of the apportionment issue. On the one hand there were too many representatives, and on the other hand there were not enough because they were now representing far more people than the 30,000 originally envisioned. The 1910 Apportionment Act had set the house at 433.
The people that were in favor of fixing the number were in part drawn from rural parts of the country that feared losing out on their proportion of power to the fastly urbanized parts of the country. Those in favor of fixing the number eventually won out so they passed the Apportionment Act of 1929 which fixed the number at 435.
If you are interested there is an excellent layout of the various methods that have been employed for reapportionment by Congresses throughout the years: Methods of Apportionment.
One factor I don't see mentioned, but which has to be seen as a limiting factor, is the size of the physical US Capitol building. It is a historic building, having been built in the early 19th Century and expanded to (mostly) its current state in the mid 19th. The House Chamber only has 448 permanent seats. I suppose you could start sitting extra representatives in the gallery, but that would remove that seating area from public use (which would be a crying shame). So expanding the number of representatives much past the current 435 would require an extensive remodeling, or worse yet, a larger chamber.
There is so much history in that Chamber that frankly, tearing it down to build a new one would be a crime against history.