Can a human cell live indefinetely in a controlled environment?
This question is at least two questions. 

Dividing cells

In terms of a dividing human cell line, every time a division occurs the telomeres capping the ends of the chromosomes get a little bit shorter. Once the telomeres get short enough they act as a signal that triggers apoptosis, destroying the cell. There is some human-to-human variation in the initial length of the telomeres, but the impression I've gotten from hanging around cell biologists is that the division limit is somewhere on the order of 10-100.

There are also cells in which the activity of a protein called telomerase is upregulated. Telomerase is a type of polymerase (an enzyme that synthesizes DNA) that acts to add extra length to telomeres. Telomerase+ cells are not affected by the division limit in the normal way. Some examples of telomerase+ cells are stem cells and many kinds of cancer cells. 

Stationary cells

In a very real sense, a cell "dies" when it divides. Upon division a cell splits into two daughter cells, but it cannot be said that either daughter cell "is" the original cell. So a more satisfying way to answer to the original question would be to look at how long a single cell can be kept as-is and recognizably alive (i.e. in stationary phase).

If you cheat and freeze the cell at liquid nitrogen temperatures (78 K), then there are plenty of examples of cells surviving for decades, in the sense that a culture can be stored and then thawed, and normal life processes will resume at least some of the time.

Sadly, there may not be a proper answer out there to this version of the question. A quick lit search did not turn up much. It is thought that at least some cells (in particular neurons) last for decades in the human body without dividing, although this is unsurprisingly hard to prove.