First off, I'm not a historian. I am, instead, an amateur who has an absolute obsession for all things Rome.

Rome being a slave-holding society, it's hard to miss this aspect. I recommend the book The Roman Guide to Slave Management: A Treatise by Nobleman Marcus Sidonius Falx by Jerry Toner. You'll notice the two "by" mentions. It's written by Jerry Toner, a modern historian, but he cleverly created a "typical" Roman aristocrat named Marcus Sidonius Falx as a point of view character to help get the ancient Roman aristocratic perspective across. Unfortunately, we don't have sources for the slave perspective. It's a well researched book and the basis for my answer below. If you do read it, also invest in a book cover if you want to read it in public. The name is very... eye catching.

Unfortunately, there is no single answer to your question. Masters differed in their treatment of slaves both between different time periods and even during the same time period.

It does seem, at least from a general look at the material, that the treatment of slaves became marginally better over time. At least there became a few laws to protect them, and thought leaders of the day seemed to become more predisposed to treating their slaves with a modicum of decency.

Cato was a prominent politician who lived during the time of the Republic, before the Empire. He apparently just stopped feeding his slaves who got too old or sick to work (I got this from the book mentioned above, but the original reference is apparently in Plutarch's Life of Cator the Elder). I'm not sure if this was a widely held opinion - I really hope not - but it does demonstrate that at least one thought leader of the time felt it was acceptable to do this.

The story shifts slightly once we get to the Empire. The Emperor Claudius, for instance, made an edict to stop people from dumping sick or old slaves on the Tiber Island.

"When certain men were exposing their sick and worn out slaves on the Island of Aesculapius​ because of the trouble of treating them, Claudius decreed that all such slaves were free, and that if they recovered, they should not return to the control of their master; but if anyone preferred to kill such a slave rather than to abandon him, he was liable to the charge of murder." - Suetonius

This does suggest that during that time people were doing this to get rid of slaves they didn't want to care for. Perhaps Cato's views were more widely accepted. Or perhaps a few bad masters were causing a public nuisance at the temple. It's hard to say, but to me it suggests two things: 1) some people were really, really mean, at least as we'd see it today and 2) there were some people in ancient Rome who saw this as reprehensible behavior, though probably more as a public nuisance than as a human rights issue.

Meanwhile, Seneca, who lives around the same time as Claudius above, tells of a story where an old slave who was too old and worn-out to work anymore was given the easy job of door guard. Seneca argued for greater care to be given to slaves, though, so he might have been representing the far side of this spectrum, rather than the consensus of the greater majority. Also, it is worth noting that this old slave was an old childhood playmate of his master. He wasn't just an old man, he was an old friend, and that personal relationship may have warranted the kinder treatment (though of course, his "friend" didn't free him, but there you go. Rome).

I do also suspect that some masters just freed their slaves as they got too old, after a term of service, so that way they didn't have to care for them, but I didn't read that in my book. This would be different than just freeing a slave when they got too old to do anything, though, since there was still a link between a master and his now-freedman. The freedman could be a useful agent for the master in business dealings, as well as help the master seem impressive through a particular type of Roman conspicuous consumption - being seen to be so rich you weren't just surrounded by a bunch of slaves, but also a bunch of freedmen. You were so rich you could grant freedom to so many.

I hope this response demonstrates at least the range of potential outcomes for a slave in the Roman period - including the potential for public attitudes to shift over time.

As I am not a historian and haven't done a full study on the topic, I'm looking forward to what others answer here!