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The bias of TCP's congestion avoidance mechanism against connections with long round trip times (RTT) is a known fact. Many alternative congestion avoidance policies have been proposed to improve the fairness. Though the proposed policies attempt to address and resolve the fairness issue, we show that they tend to be harmful to connections that traverse either slow links, like 56 Kbps modem links, or long thin networks (LTN), like cellular links. We specifically consider a very common scenario where the last-hop link connecting the end user (i.e., client) may be a slow link or LTN. In this case, the TCP sender (i.e., server) is usually unaware of the network path, and when it is equipped with such policies, it increases probing into the network in a quest for the non-existent bandwidth. We conduct simulation studies to evaluate the impact of the proposed policies on connections that traverse either slow links or LTNs. We notice that the proposed policies cause increased buffer overflows at the last-hop router, thereby degrading the performance of the connection. We study the impact of increased buffer sizes at the last-hop router and also the effect of advertising a limited receive window. We show that the impact of the policies, on connections traversing slow links or LTNs, can be reduced by selectively disabling the policies.