Who has a security system and dialup?

The "ringer equivalency" of all security panels is "0" (zero). When it seizes the phone line your modem won't just "slow down", it will stop communicating all together. The problem with your sister's line may have to do with connections. I would suggest checking out the whole house starting at the protector. Remove the wires from the terminal lugs and (if you have sufficient length) cut and re-strip them. Clean off any corrosion from the terminal lugs with a small wire brush. Wrap the newly stripped wires around the posts and tighten the nuts/screws. If the protector is exposed to the elements (subject to corrosion) you might consider applying some petroleum jelly (or similar non-corrosive lubricant) to the screw terminals after you've tightened them down.

Continue on inside the house and remove each of the phone jacks from the wall. Restrip the wires and tighten all the screws. Check to make sure the pins inside the connector are making solid contact with the phone plug. A loose connection will degrade the performance (and cause "noise"). Make sure the phone cables to the modem aren't routed around (or near) any high voltage devices. Last but not least, check the "ringer equivalency" of all the household phones. Add all of them together and if the total exceeds 2.0, consider unplugging one or two of them. If it exceeds 4.0, you may have problems getting your phones to ring properly.