Short answer: Based on your description of the scenario, it's possible, but seemingly unlikely, that interference received from the ham radio is causing your problems.
Kevin's answer is good (especially his recommendation to talk to your neighbor and get his input,) but I just wanted to add a few things regarding the technical aspects involved in answering your question:
General Background on RF Communications and Hams
Most forms of home Internet connections use RF (radio-frequency) signals to transmit the data, regardless of whether that's over a wire or over the air. This is true for DSL, cable, and any currently-popular form of wireless ISP. The only major exception to this is fiber-optic ISPs, which use light signals instead of RF signals (but even those could potentially have RF equipment in the transceivers, that is, in the modem.)
Additionally, home networks use RF signals to transmit information (i.e. Wi-Fi, Bluetooth, and even wired Ethernet.)
All forms of RF communication, whether wired or wireless, operate in some frequency band. In most cases, only signals (whether intentional signals or unintentional ones like harmonics and spurs) in or near the operating band will cause interference with a given communication system. The frequencies of these bands are measured in Hertz (Hz.) 1 kHz = 1,000 Hz, 1 MHz = 1,000,000 Hz, and 1 GHz = 1,000,000,000 Hz.
Ham radio operators broadcast signals on frequencies ranging from around 135 kHz up to around 30 MHz. This is relatively low compared to most modern RF communication.
Potential Effects on Different Types of Internet Connections
Here's a rough guide to how different networking and Internet connection technologies work, including which frequency bands they use and how they could be affected by a ham operator:
DSL ISPs typically use quite low frequencies, in the kHz to the tens of MHz. Notice that this is the same general region of the spectrum used by ham operators. Furthermore, DSL operates on normal home phone cables, which are not very well shielded because they were originally only designed to carry baseband voice signals which only go up to a bit over 3 kHz. Because of this poor shielding, all of the phone cables in your home can act as a big antenna and receive any radio signals that are being transmitted over the air nearby at high enough power levels.
As such, it's actually quite possible for a ham operator who is currently transmitting enough power in the direction of your home to interfere with a DSL signal enough to cause high packet drop rates or even total connection loss, especially if your phone lines and/or DSL modem aren't shielded well (which they probably aren't.)
Cable ISPs typically use much higher frequencies in the hundreds of MHz up into the GHz. These signals are then transmitted down a coaxial ('coax') cable that is usually shielded pretty well. Since the cables are shielded reasonably and the frequency bands are quite far from those used by hams, it's very unlikely that a ham's operations could interfere with a cable Internet connection unless the ham radio was sitting right beside the cable and/or modem and transmitting a high amount of power.
Wi-Fi, Cellular Data Networks, et al.
Wi-Fi, Bluetooth, and other relatively short-range data networks generally use either the 900 MHz band, the 2.4 GHz band, or the 5.8 GHz band (and some newer equipment uses 60 GHz.) Cellular data networks generally operate from around 800 MHz to around 2.2 GHz. As such, these networking technologies all operate in frequency bands quite far from where hams broadcast. That said, unlike wired technologies, these technologies are intentionally designed to receive signals that are broadcast over the air. Due to the short length of their antennas and their receiver designs, they will not pick up signals in the ham bands well, but they will pick them up somewhat if there's enough power being broadcast.
As a result of these factors, these technologies will be significantly more susceptible to interference from a ham transmission than a cable connection would be, but probably not as susceptible as a DSL connection. A ham radio transmitting close to a Wi-Fi station could take it down, but probably not one at your neighbor's house unless he's really aiming a lot of power at you.
As other answers have suggested, it's quite possible for ham radio transmissions to interfere with a DSL signal. If your Internet connection is cable, though, then the tech who suggested the ham might be the problem was probably smoking something. That said, the patterns you've described for the behavior of your Internet connection suggest that, even if you are using DSL, it's more likely a problem with your modem or ISP, as it's unlikely that your neighbor is transmitting all day long.
If you do use DSL, there are a couple of ways to go about confirming or denying whether your neighbor's ham radio is the source of your problems:
Talk to Your Neighbor
As others have suggested, talk to him. He'll probably tell you when he's been transmitting and the patterns of his transmit times either will or won't line up with when you've been having problems.
Use a Spectrum Analyzer
Another possibility is to use a spectrum analyzer to check out the signal integrity and look for possible interfering signals. Honestly, the tech who visited your house really should have already done this in attempt to diagnose the problem. The cable techs who have been to my house in the past have indeed hooked the line up to a handheld SpecAn to check out the signal. If you have a friend who happens to be an electrical engineer (especially an RF engineer) or an RF or ISP tech, they might be willing to bring one over and hook it up to your phone lines to see what the relevant portion of the spectrum looks like. This should pretty definitively tell you whether the problem is due to an interfering signal or not. If it is due to an interfering signal, you could hook the SpecAn up to an antenna designed for the appropriate band and see if the problem signal gets stronger as you go toward your neighbor's antenna.