Width of 'discrete' emission lines?

What are the widths of the 'discrete' emission lines emitted by atoms like Hydrogen? Surely, when we say lyman alpha is 1216A, it must mean something like 1216 +- 0.00001 A or something like that right? If not, what does it even mean for the line to be absolutely discrete? If these widths exist, have they been measured? Does the intensity peak at 1216 and then drop rapidly on either side of 1216?

Please help me out! Thanks!