One thing that everyone needs to take into consideration is: When you connect to the Steam server, it is not a direct connection from your computer to the Steam server. The connection goes through many other servers along the way. A dropped connection or slow connection can happen at anyone of those servers, causing the update to not complete. Hence the suggestion to do the Verify thing more than once to make sure all your files are up to date.
Here's an example of the connection from my computer in north Georgia to the Steam store main page. The ms on each entry is milliseconds.
Tracing route to store.steampowered.com [208.64.202.69]
over a maximum of 30 hops:
1 * * * Request timed out.
2 10 ms 10 ms 10 ms xe-10-0-0-sur01.y2cartersvil.ga.atlanta.comcast.net [68.85.232.169]
3 14 ms 10 ms 10 ms xe-1-0-1-0-ar01.b0atlanta.ga.atlanta.comcast.net [68.85.109.57]
4 14 ms 14 ms 14 ms pos-3-8-0-0-cr01.atlanta.ga.ibone.comcast.net [68.86.95.197]
5 35 ms 34 ms 34 ms pos-1-7-0-0-cr01.dallas.tx.ibone.comcast.net [68.86.85.149]
6 54 ms 79 ms 79 ms pos-2-11-0-0-cr01.denver.co.ibone.comcast.net [68.86.87.29]
7 83 ms 104 ms 104 ms pos-0-11-0-0-cr01.seattle.wa.ibone.comcast.net [68.86.88.6]
8 83 ms 104 ms 79 ms be-10-pe03.seattle.wa.ibone.comcast.net [68.86.84.74]
9 83 ms 104 ms 79 ms as32590.seattle.wa.ibone.comcast.net [66.208.233.26]
10 83 ms 104 ms 79 ms 34.6.196.205.in-addr.arpa [205.196.6.34]
11 83 ms 104 ms 104 ms fw01-core.tuk.valve.net [208.64.202.21]
12 81 ms 104 ms 104 ms 208-64-202-69.valve.net [208.64.202.69]
Trace complete.
The number of servers your individual connection has to go through (
hops) depends on your location and the location of the server your connecting to. It could be less or more than shown above, and any server in the list could have a problem, causing a problem on your connection.