I experienced this as well. Last week, I restored my phone and since then, I have not experienced the tone skipping when I get a text. I don't know if you want to restore or not, but it's been a week, and I haven't experienced any hiccups.
I will get this from time to time. I turn the phone off and back on and it works again. Also lately my LED Flash notification has been sporadic. I noticed it had not been working, and it had been quite some time. I cycled it and it worked. It stopped working and I turned the phone off and back on and it has been working since.
If you have the message app open and are actively texting, incoming texts won't make the usual sound. There's a kind of a soft "whooshing" sound when another text comes in, either in the conversation you have open or in another one. The notification sound will occur if you are out of the message app, doing something else or idle.