I am also getting LTE connection problems on my iphone 5. Many of at&t's LTE markets were rushed to completion right before the launch of iphone 5 and they are rushing to finish the ones they promised they said would be done by the end of the year. In doing so, at&t DID NOT upgrade all of the towers in your area to LTE. Because LTE is only used by about 5% of at&t's current users, what they have decided is to only upgrade a few towers and turn their LTE antenna's into "macro-cells" which are towers able to send signal further than normal towers. Cell phone companies normally have macro cells in rural area to get maximum coverage for least infrastructure. But since LTE has relatively low adoption rates as of now, they can get away with upping the output power on cell towers for the time being. As soon as more people in your area get onto LTE, the towers will become strained and at&t will eventually upgrade all the towers in your area and turn down the power. If you noticed, your LTE drops off at around 1-2 bars meaning you are at the edge of the LTE tower's network and when your phone detects too low of a signal it will automatically connect to the nearest tower and use its HSPA+ until it can reliably reconnect to an LTE tower. This is a way your phone can dynamically save battery power since the LTE tower may be too far.
Verizon did a better job at rolling out its LTE network by planning well into the future and building out its network early in anticipation for LTE for the masses. At&t is behind especially in locations where LTE was launched in the last 6 months.
FYI my area was launched 4 days before iphone 5 launch. my iphone jumps back and forth between lte and 4g all day and almost every where i go unless i have full strength LTE
