Cell line do change their biochemistry overtime with passage. This is especially true with primary cell lines and less with immortalized cell lines.
I personally work with NIH 3T3, HFF as my primary cell lines. For those cell line I make sure keep track of cell passage every time I split. Also, I tried to use them passage 3 to 10 for experimentation. I still passage to later passages to use as monolayer for titering virus. I also observed that cell with passage over 19 or above ( also split at 90% confluencent) loose the ability to adhere to flask, some other time they loose contact inhibition, etc.
immortalized cell lines like Hela are fine despite being passage hundred of times ( at least in observe physico biochemical characteristics. )
I kept passage low by split T175 flasks 1 to 6 or 1 to 8 at early passage at let them grow to 90% confluent. Then each flask is deposit into 10 vials of cryogenic tubes.
As always, it all depends on the scientific question you want to address.
For basic research it may be totally acceptable to work with standard cell lines in passage number unknown + 255 or similar. Since basic signaling or biochemical properties may still be the same as in a normal or a tumor cell in vivo.
However, if you are for examples (as my lab is) looking for drug responses or resistance development of tumors, only cells in low passages are acceptable - and there are many peers that doubt even that ;-)
The best solution for most purposes is to initially make stocks of the cell line of interest in the lowest passage number possible (and do also perform QC like mycoplasma, bacteria and virus exclusion) and subsequently work with defined passage numbers. This will at least most likely allow for a safe reproduction of results.
We establish primary cell lines from patients and usually work with them to an absolute max. of passage number 50.