This is feeling like a hardware issue. Definitely not on Apache's end or anything. Compare the ping patterns:
Stevie:
ashoat$ ping 65.19.143.2
PING 65.19.143.2 (65.19.143.2): 56 data bytes
64 bytes from 65.19.143.2: icmp_seq=0 ttl=54 time=8.674 ms
64 bytes from 65.19.143.2: icmp_seq=1 ttl=54 time=4.040 ms
64 bytes from 65.19.143.2: icmp_seq=2 ttl=54 time=6.876 ms
64 bytes from 65.19.143.2: icmp_seq=3 ttl=54 time=6.753 ms
64 bytes from 65.19.143.2: icmp_seq=4 ttl=54 time=5.171 ms
64 bytes from 65.19.143.2: icmp_seq=5 ttl=54 time=6.472 ms
64 bytes from 65.19.143.2: icmp_seq=6 ttl=54 time=6.444 ms
64 bytes from 65.19.143.2: icmp_seq=7 ttl=54 time=3.401 ms
64 bytes from 65.19.143.2: icmp_seq=8 ttl=54 time=4.694 ms
64 bytes from 65.19.143.2: icmp_seq=9 ttl=54 time=8.337 ms
Request timeout for icmp_seq 10
64 bytes from 65.19.143.2: icmp_seq=11 ttl=54 time=42.790 ms
Request timeout for icmp_seq 12
Request timeout for icmp_seq 13
Request timeout for icmp_seq 14
Request timeout for icmp_seq 15
Request timeout for icmp_seq 16
Request timeout for icmp_seq 17
Request timeout for icmp_seq 18
Request timeout for icmp_seq 19
64 bytes from 65.19.143.2: icmp_seq=20 ttl=54 time=51.253 ms
Request timeout for icmp_seq 21
Request timeout for icmp_seq 22
Request timeout for icmp_seq 23
Request timeout for icmp_seq 24
Request timeout for icmp_seq 25
Request timeout for icmp_seq 26
64 bytes from 65.19.143.2: icmp_seq=27 ttl=54 time=46.454 ms
64 bytes from 65.19.143.2: icmp_seq=28 ttl=54 time=42.953 ms
Request timeout for icmp_seq 29
64 bytes from 65.19.143.2: icmp_seq=30 ttl=54 time=45.583 ms
64 bytes from 65.19.143.2: icmp_seq=31 ttl=54 time=41.809 ms
Request timeout for icmp_seq 32
64 bytes from 65.19.143.2: icmp_seq=33 ttl=54 time=44.662 ms
64 bytes from 65.19.143.2: icmp_seq=34 ttl=54 time=43.495 ms
Request timeout for icmp_seq 35
Request timeout for icmp_seq 36
64 bytes from 65.19.143.2: icmp_seq=37 ttl=54 time=42.187 ms
Request timeout for icmp_seq 38
64 bytes from 65.19.143.2: icmp_seq=39 ttl=54 time=46.252 ms
Request timeout for icmp_seq 40
Request timeout for icmp_seq 41
Request timeout for icmp_seq 42
64 bytes from 65.19.143.2: icmp_seq=43 ttl=54 time=42.497 ms
Request timeout for icmp_seq 44
Request timeout for icmp_seq 45
Request timeout for icmp_seq 46
64 bytes from 65.19.143.2: icmp_seq=47 ttl=54 time=42.923 ms
Request timeout for icmp_seq 48
64 bytes from 65.19.143.2: icmp_seq=49 ttl=54 time=44.294 ms
Request timeout for icmp_seq 50
64 bytes from 65.19.143.2: icmp_seq=51 ttl=54 time=44.736 ms
64 bytes from 65.19.143.2: icmp_seq=52 ttl=54 time=46.710 ms
Request timeout for icmp_seq 53
Request timeout for icmp_seq 54
64 bytes from 65.19.143.2: icmp_seq=55 ttl=54 time=43.471 ms
Request timeout for icmp_seq 56
64 bytes from 65.19.143.2: icmp_seq=57 ttl=54 time=44.710 ms
Request timeout for icmp_seq 58
64 bytes from 65.19.143.2: icmp_seq=59 ttl=54 time=46.111 ms
64 bytes from 65.19.143.2: icmp_seq=60 ttl=54 time=47.208 ms
Request timeout for icmp_seq 61
64 bytes from 65.19.143.2: icmp_seq=62 ttl=54 time=43.252 ms
Request timeout for icmp_seq 63
^C
--- 65.19.143.2 ping statistics ---
65 packets transmitted, 30 packets received, 53.8% packet loss
round-trip min/avg/max/stddev = 3.401/31.807/51.253/18.300 ms
Charlie:
ping 64.62.211.131
PING 64.62.211.131 (64.62.211.131): 56 data bytes
64 bytes from 64.62.211.131: icmp_seq=0 ttl=54 time=6.261 ms
64 bytes from 64.62.211.131: icmp_seq=1 ttl=54 time=7.245 ms
64 bytes from 64.62.211.131: icmp_seq=2 ttl=54 time=17.679 ms
64 bytes from 64.62.211.131: icmp_seq=3 ttl=54 time=6.886 ms
64 bytes from 64.62.211.131: icmp_seq=4 ttl=54 time=3.846 ms
64 bytes from 64.62.211.131: icmp_seq=5 ttl=54 time=5.068 ms
64 bytes from 64.62.211.131: icmp_seq=6 ttl=54 time=6.939 ms
64 bytes from 64.62.211.131: icmp_seq=7 ttl=54 time=7.360 ms
64 bytes from 64.62.211.131: icmp_seq=8 ttl=54 time=5.875 ms
64 bytes from 64.62.211.131: icmp_seq=9 ttl=54 time=6.680 ms
64 bytes from 64.62.211.131: icmp_seq=10 ttl=54 time=4.582 ms
64 bytes from 64.62.211.131: icmp_seq=11 ttl=54 time=13.068 ms
64 bytes from 64.62.211.131: icmp_seq=12 ttl=54 time=3.598 ms
64 bytes from 64.62.211.131: icmp_seq=13 ttl=54 time=6.716 ms
64 bytes from 64.62.211.131: icmp_seq=14 ttl=54 time=23.143 ms
64 bytes from 64.62.211.131: icmp_seq=15 ttl=54 time=31.943 ms
64 bytes from 64.62.211.131: icmp_seq=16 ttl=54 time=25.994 ms
64 bytes from 64.62.211.131: icmp_seq=17 ttl=54 time=56.681 ms
64 bytes from 64.62.211.131: icmp_seq=18 ttl=54 time=5.782 ms
64 bytes from 64.62.211.131: icmp_seq=19 ttl=54 time=6.376 ms
64 bytes from 64.62.211.131: icmp_seq=20 ttl=54 time=17.729 ms
64 bytes from 64.62.211.131: icmp_seq=21 ttl=54 time=4.384 ms
64 bytes from 64.62.211.131: icmp_seq=22 ttl=54 time=7.209 ms
64 bytes from 64.62.211.131: icmp_seq=23 ttl=54 time=34.841 ms
64 bytes from 64.62.211.131: icmp_seq=24 ttl=54 time=7.089 ms
64 bytes from 64.62.211.131: icmp_seq=25 ttl=54 time=5.170 ms
^C
--- 64.62.211.131 ping statistics ---
26 packets transmitted, 26 packets received, 0.0% packet loss
round-trip min/avg/max/stddev = 3.598/12.621/56.681/12.367 ms
It also doesn't seem to be the upstream router - pings there return fine. I'm guessing it might be the network card, and I'm hoping it's the patch cable. I'll be gone for most of this weekend but I'll see if I can replace the patch cable sometime early next week.