Can my company punish me for being off sick if I provide sick notes and hospital appointment letters as evidence?

Yes if you are sick excessive, Sadly a company esp in the US, look at the bottom line, they have jobs to be done, and a person who is out too much for any a good reason is still not getting their job done, often other employees may have to work over time, some one has to be called in from a day off, or work is just not done. Most companies have a maximum allowed days you can misss during a quarter or year, and if you start missing more than that, you are subject to being written up.

If you need an extended amount of time off, such as a month or two, there is the FMLA a federal program to allow you time off for extended amount of time for family or personal illness and other reason.

For example if you were sick two days a week for the last 5 months, I don't know of any boss that would keep you around.

Lol, I haven't been off any where near that. Are complny policy is 3 times in a year leads to punishment. They made me take one of them for having my tonsils out and one for another op I had this year and obviously I have had health problems to do with them.

So your company has this policy that if you take more than 3 sick days in a year it leads to punishment?
What sort of punishment?

3 times or 3 days?

3 times which is really annoying because one time I came in thinking I was better and got sent home half way through the day so it counted as 2. 3 in a year means you lose your commission for the month which is around £300-£500 plus you loose your xmas bonus which is £500. Then if your off again that year you loose your commission again. 5 can lead to being sacked. The result is that everyone was coming into work ill to aviod this and making me ill before I had my tonsils out because I had no immune system.