I think you are mixing terms here... SSIDs imply wireless networks, and yet you are talking about "splitters" and people "Wired in"? Are you on a Cable modem connection ? Are you talking about them possibly using your Wireless network?
1. Splitters are used on a coax cable. If you split a coax signal, you can get the same open TV channels, but not any paid programming these days, those channels require a cable box (one customer can rent multiple cable boxes under the same account, then it is possible for them to use the same TV service, but it is against the TOS and does not apply to the internet connection).
1a. You can't just split the coax signal and hook up another cable modem - each cable modem is provisioned by your ISP by its MAC address.
2. You CAN split an Ethernet wired network easily - Cat5 cables can be hooked to a switch, and anyone with physical access to it will be on the same network.
3. Wireless networks and SSIDs - I assume that is your question, if it is possible that they are using your wireless network? If you noticed a new SSID in the area, it is very likely that they hooked up a wireless router for their own use. If it is "open" (not requiring login?), then you should advise them to secure it using WPA2, otherwise anyone in the area can hook up to their network. A new SSID in the area does not constitute any cause of security concern, it is a common thing. It could, however introduce some interference if it is a strong signal in proximity to your wireless network. The best think to do is to make sure you use a non-overlapping channel for your Wi-Fi network (at least a couple of channels apart from theirs). You'd have to do a wireless survey to see what network is using what channels/frequencies using some software like the "WiFi Analyzer" phone app, for example.
You can usually see some type of usage meter/client list on your NAT router/gateway/modem or ISP to determine if anyone is connected/using your network. It is likely just new interference causing you issues because of overlapping channels. Sometimes simply rebooting your wireless router can fix it (if set to automatically select the "best" channel), but it is best to do the manual wireless survey and learn more about your network/devices in the process.
I hope this helps, we'd need more info (device models, ISP type, etc.) to offer more specific help.
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