I can hear it quite clearly. Might be technical issue. There is some ‘running together’ of the end of ‘rheumatismes’ and ‘me’, as occurs more markedly in fast everyday speech. 

I thought it may be the 'running together' of 'rhumatismes' and 'me' but even taking that into consideration I still can't hear it. Must be too quick for my brain to process!

Sorry to go on about this but it's really bothering me.....

Should I be hearing 2 'me' sounds before dérangent - as in "mes rhumatismes me dérangent ?

Hi G. I too can only hear the "me" sound once in "rhumatismes me dérangent", and I played it numerous times, but I wouldn't be too worried about trying to hear two. I think in conversation they would usually run the two "me" (or similar) sounds together like this, for ease of pronunciation (otherwise it would sound a bit like a stutter). In the spoken language I have found that there are often unsounded words or sounds, just like the missing "ne" in negative sentences which is almost universally dropped. Or the 'il" in "il y a" or "il faut" for example.

Thanks everyone, I'm not losing sleep over it :-) and I do understand it's all part of french pronunciation. I was more concerned that I couldn't hear what everyone else was hearing. Thanks to Paul I now know I'm not alone!! I've decided to cancel my hearing test for another year.......LOL

To my (amateur) ear the “me” is completely missing, and so Paul’s explanation seems quite convincing.

The first time I transcribed it, I put in the me because I knew it should be there.  But after listening for several times, I concluded she was speaking some kind of slang and omitting the me.  So, I took it out and found I was wrong.  I guess it's something like "I didn't know" in English.  Sometimes the nt gets swallowed and the listener has to say "did or did not?" for clarification.