Is there a limit? I couldn't find one posted.
3 Answers
Have a look at this link** and this one (and this for 2013). A quick summary:
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  Max. Rows  Max. Columns  Max. Cols by letter 
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 Excel 365*  1,048,576  16,384  XFD 
 Excel 2013  1,048,576  16,384  XFD 
 Excel 2010  1,048,576  16,384  XFD 
 Excel 2007  1,048,576  16,384  XFD 
 Excel 2003  65,536  256  IV 
 Excel 2002 (XP)  65,536  256  IV 
 Excel 2000  65,536  256  IV 
 Excel 97  65,536  256  IV 
 Excel 95  16,384  256  IV 
 Excel 5  16,384  256  IV 
+++++
*Excel 365 unverified.
**This (web archive) link probably will not work with your browser, but the information is in the page source.

16Links are apparently dead. This answer, though it does in fact provide most of the information, indicates the limitations of Excel, and not of the supported file formats xls or xlsx. The limitations of xls is still 256 columns and 65536 rows, if you use that file format in a newer Excel version. May 29, 2015 at 9:11

2@LasseV.Karlsen: Thanks for the update. It looks like the second link is still alive, but I have noted that the first link is dead. May 29, 2015 at 17:19

3MS doesn't seem to have the 2003 and before specs anymore, but on Wayback you can sort of get to them... web.archive.org/web/20080313035725/http://office.microsoft.com/… for example (I edited that in).– JoeJun 3, 2015 at 22:11
You can find out how many rows your version of Excel supports by opening a new file in Excel then pressing CTRL + Down Arrow in an empty worksheet.
This number of rows can be saved in the default format that version of Excel supports (e.g. XLSX).
However, you will not necessarily be able to save in the older XLS format, which is still limited to 65,536 rows. Typing in a cell in row number 1048576 (e.g. A1048576) then saving to XLS format will cause a warning. If you close and reopen this file, you will then find this cell empty.
Doing a web search for "maximum number of rows in Excel" gives a link (several, actually) with
Starting in Excel 2007, the "Big Grid" increases the maximum number of rows per worksheet from 65,536 to over 1 million, and the number of columns from 256 (IV) to 16,384 (XFD).