Hi good afternoon.
Yes that will happen eventually, when the wires become too old and have been tuned a few times.
You will need to have them replaced, at least the ones that wont stay in tune.
Its not expensive for the wires and they can be replaced by most experienced tuner technicians.
Sometimes though you need a piano repairman to come in and do that for you. but usually that is all that needs to be done to fix this issue and have it playing like new;-)
If you do a local web search near your area, you should be able to locate one. If not, please let me know and also let me know your state and town and I will see who I can recommend.
It sounds like it was well cared for so I think all you will need are new tone wires (strings).
In any case, I will stick with you until this is complete.
Please rate my service kindly.
Also it is the only way I am credited for my time and expertise with you.
Once you rate, I will be notified and will add your question to my priority list so you can reach me anytime until this problem is solved.
Then you can reach me by simply replying to this message anytime.
Thank you and have a great weekend!