The time that ARA will wait to establish a connection is set in the CCL (command control language) script for a particular modem. The CCL script can be modified to change that limit, and CCL script editing tools are available from APDA.
Another way to change the time limit is by including some commas to the end of the dialing string. Every comma attached to the end of the number will add approximately 2 seconds to the amount of time ARA will wait before it assumes the connection will not establish itself. Below is an example:
Number: 5551234,,,,,,,,,,
This dialing string will add approximately 20 seconds to the default time, which is usually 50 seconds.
The Tech Info Library article titled "Locating Vendor Information" can help you search for the APDA address and phone number.