Nope.
Current output of netstat -a | grep (my external address)
on my linux box using deluge and actively downloading a well-seeded ubuntu distro.
tcp 0 0 xx.xx.xx.xx:34683 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:44990 xxx.xxx.190.165:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:44977 xxx.xxx.190.165:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:52057 xxx.xxx.190.148:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:57622 xxx.xxx.188.249:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:34046 xxx.xxx.190.161:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:46295 xxx.xxx.137.202:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:60520 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:51588 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:39297 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:40050 xxx.xxx.137.216:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:45003 xxx.xxx.190.165:socks FIN_WAIT2
tcp 0 0 192.168.1.10:8112 192.168.1.2:51985 ESTABLISHED
tcp 0 0 xx.xx.xx.xx:51270 xxx.xxx.190.157:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:39284 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:44005 xxx.xxx.190.147:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:60507 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:43992 xxx.xxx.190.147:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:45815 xxx.xxx.190.169:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:51636 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:43561 .:socks FIN_WAIT2
tcp 0 0 192.168.1.10:8112 192.168.1.2:51983 ESTABLISHED
tcp 0 0 xx.xx.xx.xx:51623 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:50018 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:55134 tsn109-201-152-23:socks FIN_WAIT2
tcp 1 0 xx.xx.xx.xx:42218 einstein.phys.uwm:https CLOSE_WAIT
tcp 0 0 192.168.1.10:8112 192.168.1.2:51982 ESTABLISHED
tcp 0 0 xx.xx.xx.xx:57044 xxx.xxx.190.158:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:43310 xxx.xxx.137.205:socks FIN_WAIT2
tcp 0 0 192.168.1.10:8112 192.168.1.2:51982 ESTABLISHED
tcp 0 0 xx.xx.xx.xx:57044 xxx.xxx.190.158:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:43310 xxx.xxx.137.205:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:43561 .:socks FIN_WAIT2
tcp 0 0 192.168.1.10:8112 192.168.1.2:51983 ESTABLISHED
tcp 0 0 xx.xx.xx.xx:51623 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:50018 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:55134 tsn109-201-152-23:socks FIN_WAIT2
tcp 1 0 xx.xx.xx.xx:42218 einstein.phys.uwm:https CLOSE_WAIT
tcp 0 0 192.168.1.10:8112 192.168.1.2:51982 ESTABLISHED
tcp 0 0 xx.xx.xx.xx:57044 xxx.xxx.190.158:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:43310 xxx.xxx.137.205:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:45016 xxx.xxx.190.165:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:55827 xxx.xxx.137.207:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:52131 tsn109-201-152-11:socks TIME_WAIT
tcp 0 0 xx.xx.xx.xx:51302 xxx.xxx.190.157:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:52410 xxx.xxx.190.140:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:51675 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:39831 xxx.xxx.190.139:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:44945 xxx.xxx.190.165:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:41724 xxx.xxx.190.155:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:43611 xxx.xxx.190.145:socks FIN_WAIT2
tcp 0 0 192.168.1.10:8112 192.168.1.2:51984 ESTABLISHED
tcp 0 0 xx.xx.xx.xx:52190 xxx.xxx.137.206:socks TIME_WAIT
tcp 0 0 xx.xx.xx.xx:51662 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:51601 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:50044 .:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:57710 tsn46-166-168-209:socks FIN_WAIT2
tcp 0 0 xx.xx.xx.xx:57710 tsn46-166-168-209:socks FIN_WAIT2
...and it doesn't ever grow beyond 200 - because Deluge is sensible, enforcing the connection limit set in its options and closing old connections.