Yes they offer support, but many not offer a fix. A few years back we ran into a problem with a DES encryption bug for which a trivial fix was well known. Microsoft documented it and reported the resolution as will not fix. We had to disable the encryption bug fix to allow Internet Explorer to work.
I had one product with support where we had several bug reports closed with the response that the problem would be fixed in the next release. I finally asked when the next release was due. The response was "We aren't planning to ever to another release. I terminated support. If I had got this response for an open-source project, I likely could have fixed the problems.
Another project, we often got asked to report what the solution was when we found it. "So we will both know the solution." Support did get better over time, and they did offer solutions. In one case, I got to talk to the developer of some code we hoped to get a database handle from.
I have also dealt with support to get known patches released immediately when software was broken. In another case with the same supplier, I had to send a multi-threading problem back twice. The first fix reduced the frequency of the problem significantly, and the second appeared to resolve it entirely. Both fixes were delivered outside the release cycle.
Many open-source projects are available with paid support. This is one of the ways the projects get funded. This is in addition to the often excellent unpaid support that is often available. In tricky cases, it helps to see what the code is doing in your organization. With open-source this is relatively easy to determine.
EDIT: Most of the open-source software I use works so well I haven't needed support. A few of those where I have needed help include MySQL, Apache, Ubuntu, and Firefox. In almost all cases I have been able to get my answers from the support documentation and forums. I generally find that patches when required are made available very quickly.