Point to Point Connection Between Two Hosts

I am trying to set up a point to point connection between two Linux host machines. I've connected both of them with crossover cable, but they can't ping each other. Here are the configurations for each laptop:

Laptop 1

(eth0 configuration)
IP Address - 128.110.1.6
Subnet Mask - 255.255.255.240
Gateway - 128.110.1.1

Laptop 2

(eth0 configuration)
IP address - 128.110.1.1
Subnet Mask - 255.255.255.240
Gateway - 128.110.1.6

Is there something that I'm missing here? How can I establish connectivity between the two laptops?

Edit-1: I ran the 'arp -a' command and received the following output:

Laptop 1

? (128.110.1.1) at <incomplete> on eth0

Laptop 2

? (128.110.1.6) at <incomplete> on eth0

I'm guessing these entries in the ARP table might be due to the failed pings.

Edit-2: I discovered that one of the laptops has a bridge enabled which was previously used with a usb-to-ethernet adapter interface as eth1 and the built in ethernet interface is eth0. Not sure if that would affect anything though. Following the suggestion to run ethtool produced the following output:

Laptop 1

[root@link_3 doctor]# /sbin/ethtool eth0
Settings for eth0:
        Link detected: yes
[root@link_3 doctor]# /sbin/ethtool eth1
Settings for eth1:
        Supported ports: [ TP MII ]
        Supported link modes:   10baseT/Half 10baseT/Full 
                                100baseT/Half 100baseT/Full 
        Supports auto-negotiation: Yes
        Advertised link modes:  Not reported
        Advertised auto-negotiation: No
        Speed: 10Mb/s
        Duplex: Half
        Port: Twisted Pair
        PHYAD: 0
        Transceiver: internal
        Auto-negotiation: on
        Link detected: no

Laptop 2

[root@hitl3 doctor]# /sbin/ethtool eth0
Settings for eth0:
        Supported ports: [ MII ]
        Supported link modes:   10baseT/Half 10baseT/Full 
                                100baseT/Half 100baseT/Full 
                                1000baseT/Half 1000baseT/Full 
        Supports auto-negotiation: Yes
        Advertised link modes:  10baseT/Half 10baseT/Full 
                                100baseT/Half 100baseT/Full 
                                1000baseT/Half 1000baseT/Full 
        Advertised auto-negotiation: Yes
        Speed: 1000Mb/s
        Duplex: Full
        Port: Twisted Pair
        PHYAD: 1
        Transceiver: internal
        Auto-negotiation: on
        Supports Wake-on: g
        Wake-on: d
        Current message level: 0x000000ff (255)
        Link detected: yes 

Ensure that at least one of the Ethernet ports is GigE. Otherwise you might need a crossover cable.