The primary factor affecting connect speed is the quality of the phone circuit between your location and ours. Some parts of town have better phone lines than others. Also, phone lines are affected by weather conditions. Modems are also affected by temperature. A modem may not perform the best when too hot or too cold.
Many customers report problems when they first turn on their computers or external modems.
Finally, there can be problems when mixing different brands of modems. I have a 56K modem but can't connect above 28.8. Make sure your k56flex modem has the latest revision of k56flex code. This is usually indicated in the response to an ATI3 or ATI7 command. The response to the ATI3 or ATI7 should give the information.