In the United States, you are guaranteed a right to speedy trial by a jury of your peers through the 6th Amendment. However, because the court system is so clogged with cases in the modern United States, "speedy" is a relative term. Often times, you may wait for a year or more before your case can come to trial, even though that is the next available court date when you were first arraigned. So the government is meeting the letter of the amendment, if not the spirit.
The Patriot Act, passed after the 9/11 attacks, also allows the US government to arrest and detain any individual, citizen or not, without due process, which means no access to an attorney, no formal charge, not even public notification of detention are required. While this power has rarely been used against US citizens (that we know of), it is nevertheless a clear violation of the 6th Amendment's right to a speedy trial. Some suspects have been held for over nine years in Guantanamo Bay's Camp X-Ray Prison without even being charged.
The right to a speedy trial is generally violated in countries where the government is not really interested in protecting the rights of the accused.
The most common way to violate this right is to hold a person in jail without providing them a hearing. This very rarely happens in America because of the strength of habeas corpus protections in this country. The only time I can think of where it has been systematically violated (you can argue) is with the terrorism suspects in places like Guantanamo.
There are individual cases where people have claimed that their right to a speedy trial has been violated. Courts decide these on a case-by-case basis and there are no hard-and-fast rules about when the right has been violated. One rule of thumb, though, is that a one-year wait is presumed to be too long.